AN 


INTRODUCTION 

TO 

ALGEBRA, 

BEING    THE 

FIRST    PART 

OF  A 

COURSE   OF    MATHEMATICS, 

ADAPTED 

TO  THE  METHOD  OF  INSTRUCTION  ' 

IN  THE 
AMERICAN    COLLEGES. 


BY  JEREMIAH  DAY,  D.  D.  LL.  D. 

PRESIDENT  OF  TALE  COLLEGE. 


TWENTY  FIRST  EDITION. 


NEW  HAVEN: 


PUBLISHED    AND    SOLD    BY    KEZEKIAH    HOME    &    CO. 

Sold  also  by  Hekrick  &  Noves,  jXeu.- Haven;  N.  &  ^71)7 T)  ^^vlZ'^' 
Lord  &  Co.,  A^ew  York;   Perkins,  Marvin  &  Co.,  Boston,  F.  J.  Hunt 
ZNGTON,  Hartford  ;  Hogan  &  Thompson  and  Grigg  &  ^^^l^^J^^  Phi- 
ladelphia ;  Carey,   Ua^v  ^  Co.,  Baltimore ;  Trumax&  Smith, 
Cincinnati;  and  S.  Babcock  &  Co.  Cnarlesion,  S.  C. 

8  6  3  6     7   *  1836. 


Entered  according  to  the  Act  of  Congress,  in  the  year  1831,  by  Jeremiah  Day, 
in  the  Clerk's  office,  of  the  District  Court  of  Connecticut. 


Printed  by  Hezekiab  Howe  &  Co.  Stereotyped  by  Win.  Hagar  «fe  Co.,  N.  York. 


Mathematical 
ScreRces        Q^  /^ 


Ubraiy 


PREFACE. 


The  following  summary  view  of  the  first  principles  of  al- 
~  gebra  is  intended  to  be  accommodated  to  the  method  of  in- 
^    struction  generally  adopted  in  the  American  colleges. 

The  books  which  have  been  published  in  Great  Britain  on 
N  mathematical  subjects,  are  principally  of  two  classes. — One 
S  consists  of  extended  treatises,  which  enter  into  a  thorough  in- 
»  vestigation  of  the  particular  departments  which  are  the  ob- 
H  jects1)f  their  inquiry.  Many  of  these  are  excellent  in  their 
'^  kind  ;  but  they  are  too  voluminous  for  the  use  of  the  body 
of  students  in  a  college. 

The  other  class  are  expressly  intended  for  begmners  ;  but 
many  of  them  are  written  in  so  concise  a  manner,  that  im- 
!5  portant  proofs  and  illustrations  are  excluded.  They  are 
-  mere  text-books,  containing  only  the  outlines  of  subjects 
oo  which  are  to  be  explained  and  enlarged  upon,  by  the  pro- 
"^  fessor  in  his  lecture  room,  or  by  the  private  tutor  in  his 
«    chamber. 

5        In  the  colleges  in  this  country,  there  is  generally  put  mto 

the  hands  of  a  class,  a  book  from  which  they  are  expected  oj 

themselves  to  acquire  the  principles  of  the  science  to  which 

they  are  attending  :  receiving,  however,  from  their  instructor, 

>   any  additional  assistance  which  may  be  found  necessary.    An 

3   elementary  work  for  such  a  purpose,  ought  evidently  to  con- 

^   tain  the  explanations  which  are  requisite,  to  bring  the  sub- 

tr   jects  treated  of  within  the  comprehension  of   the  body  of 

^   the  class. 

If  the  design  of  studying  the. mathematics  were  merely  to 
^  obtain  such  a  knowledge  of  the  practical  parts,  as  is  required 
^  for  transacting  business  ;  it  might  be  sufficient  to  commit  to 
memory  some  of  the  principal  rules,  and  to  make  the  opera- 
tions famihar,  by  attending  to  the  examples.  In  this  me- 
chanical way,  the  accomitant,  the  navigator,  and  tjie  land 
surveyor,  may  be  qualified  for  their  respective  employments, 
with  very  little  knowledge  of  the  principles  that  lie  at  the 
foundation  of  the  calculations  which  they  are  to  make. 

But  a  higher  object  is  proposed,  in  the  case  of  those  who 
are  acquiring  a  liberal  education.     The  main  design  should 


Jio3590 


iv  PREFACE. 

be  to  call  into  exercise,  to  discipline,  and  to  invigorate  the 
powei^  of  the  mind.  It  is  the  logic  of  the  mathematics  which 
constitutes  tlieir  principal  value,  as  a  part  of  a  course  of  col- 
legiate instruction.  The  time  and  attention  devoted  to  them, 
is  for  the  purpose  of  forming  ^und  reasoners,  rather  than  ex- 
pert mathematicians.  To  accomplish  this  object  it  is  neces- 
sary that  the  principles  be  clearly  explained  and  demonstra- 
ted, and  that  the  several  parts  be  arranged  in  such  a  manner, 
as  to  show  the  dependence  of  one  upon  another.  The  whole 
shoidd  be  so  conducted,  as  to  keep  the  reasoning  powers  in 
continual  exercise,  witliout  greatly  fatiguing  them.  No 
other  subject  affords  a  better  opportunity  for  exemplifying  the 
rules  of  correct  thinking.  A  more  finished  specimen  of  clear 
and  exact  logic  Jias,  perhaps,  never  been  produced,  than  the 
Elements  of  Geom.etry  by  Euclid. 

It  may  be  thought,  by  some,  to  be  unwise  to  form  our  gen- 
eral habits  of  arguing,  on  the  model  of  a  science  in  which 
the  inquiries  are  accompanied  with  absolute  certainty ;  while 
the  common  business  of  life  must  be  conducted  upon  pyvbable 
evidence,  and  not  upon  principles  which  admit  of  complete 
demonstration.  There  would  be  weight  in  this  objection,  if 
the  attention  were  confined  to  the  pure  mathematics.  But 
when  these  are  connected  with  the  physical  sciences,  astro- 
nom}^,  chemistry,  and  natural  philosophy,  the  mind  has  op- 
portunity to  exercise  its  judgment  upon  all  the  various  de- 
grees of  probability  which  occur  in  the  concerns  of  life. 

So  far  as  it  is  desirable  to  form  a  taste  for  mathematical 
studies,  it  is  important  that  the  books  by  which  the  student  is 
first  introduced  to  an  acquaintance  with  these  subjects,  should 
not  be  rendered  obscure  and  forbidding  by  their  conciseness. 
Here  is  no  opportunity  to  awaken  interest,  by  rhetorical  ele- 
gance, by  exciting  the  passions,  or  by  presenting  images  to 
the  imaginatioti.  The  beauty  of  the  mathematics  depends 
on  the  distinctness  of  the  objects  of  inquiry,  the  symmetry  of 
their  relations,  the  luminous  nature  of  the  arguments,  and  the 
certainty  of  the  conclusions.  But  how  is  this  beauty  to  be 
perceived,  in  a  work  which  is  s^  much  abridged,  that  the 
chain  of  reasoning  is  often  interrupted,  important  demonstra- 
tions omitted,  and  the  transitions  tToni  one  subject  to  anotjier 
so  abrupt,  as  to  keep  tiieir  connections  and  dependej»cies  out 
of  view  1 

It  may  not  be  necessary  to  state  every  proposition  and  its 
proof,  witli  all  the  formality  whicfi  is  so  strictly  adhered  to 
by  Euclid  ;  as  it  is  not  es-sential  to  a  logical  argument,  that 


PREFACE.  V 

it  be  expressed  in  regular  and  entire  syllogisms.  A  step  of 
a  demonstration  may  be  safely  omitted,  when  it  is  so  simple 
and  obvious,  that  no  one  possessing  a  moderate  acquaintance 
with  the  subject,  could  fail  to  supply  it  for  himself.  But  thi^ 
liberty  of  omission  ought  not  to  be  extended  to  cases  in 
which  it  will  occasion  obscurit}^  and  embarrassment.  If  it 
be  desirable  to  give  opportunity  for  the  mind  to  display  and 
enlarge  its  powers,  by  surmounting  obstacles;  full  scope 
may  be  found  for  this  kind  of  exercise,  especially  in  the 
higher  branches  of  the  Mathematics,  from  difficulties  which 
will  unavoidably  occur,  without  creating  new  ones  for  the 
sake  of  perplexing. 

Algebra  requires  to  be  treated  in  a  more  plain  and  diffuse 
manner,  than  some  other  parts  of  the  mathematics;  because 
it  is  to  be  attended  to,  early  in  the  course,  while  the  mind  of 
the  learner  has  not  been  habituated  to  a  mode  of  thinking  so 
abstract,  as  that  which  will  now  become  necessary.  He  has 
also  a  new  language  to  learn,  at  tlie  same  time  he  is  settling 
the  principles  upon  which  his  future  inquiries  are  to  be  con- 
ducted. These  principles  ought  to  be  established,  in  the 
most  clear  and  satisfactory  manner  wliich  the  nature  of  the 
case  will  aduiit  of.  A  Igebra  and  geometry  may  be  consider- 
ed as  lying  at  the  foundation  of  the  succeeding  branches  of 
the  mathematics,  both  pure  and  mixed.  Euclid  and  others 
have  given  to  the  geometrical  part  a  degree  of  clearness  and 
precision  which  would  be  very  desirable,  but  is  hardly  to  bo 
expected,  in  algebra. 

-^For  the  reasons  which  liave  been  mentioned,  the  mannrer 
in  which  the  following  pages  are  written,  is  not  the  most 
concise.  But  the  work  is  necessarily  limited  in  extent  of 
subject.  It  is  far  from  being  a  complete  treatise  of  algebra. 
It  is  merely  an  introduction.  It  is  intended  to  contain  as 
much  matter,  as  the  student  at  college  can  attend  to,  with 
advantage,  during  the  sl\ort  time  allotted  to  this  particular 
study.  Theie  is  generally  but  a  small  portion  of  a  class, 
who  have  either  leisure  or  inclination,  to  pursue  mathemati- 
cal inquiries  much  farther  than  is  necessary  to  maintain  an 
honorable  standing  in  the  institution  of  which  they  are 
members.  Those  few  who  have  an  unrisual  taste  for  this 
science,  and  aim  to  become  adepts  in  it,  ought  to  be  refer- 
red to  separate  and  complete  treatises,  on  the  different 
branches.  No  one  who  wishes  to  be  thoroughly  versed  in 
niathematics,  should  look  to  compendiuiris  and  elementary 
books  for  any  thing  more  than  the  fust  principles.     As  soon 


Vi  PREFACE. 

as  these  are  acquired,  he  should  be  guided  in  his  inquiries  by 
the  genius  and  spirit  of  original  authors. 

In  the  selection  of  materials,  those  articles  have  been 
taken  which  have  a  practical  application,  and  which  are  pre- 
paratory to  succeeding  parts  of  the  mathematics,  philosophy, 
and  astronomy.  The  object  has  not  been  to  introduce  ori- 
ginal matter.  In  the  mathematics,  which  have  been  cultiva- 
ted with  success  from  the  da3^s  of  Pythagoras,  and  in  which 
the  principles  already  established  are  sufficient  to  occup)^  the 
most  active  mind  for  years,  the  parts  to  which  the  student 
ought  first  to  attend,  are  not  those  recently  discovered.  Free 
use  has  been  made  of  the  works  of  Newton,  Maclaurin, 
Saunderson,  Simpson,  Euler,  Emerson,  Lacroix,  and  others, 
but  in  a  way  that  rendered  it  inconvenient  to  refer  to  them, 
in  particular  instances.  The  proper  field  for  the  display  of 
mathematical  genius,  is  in  the  region  of  invention.  But 
what  is  requisite  for  an  elementary  work,  is  to  collect,  ar- 
range and  illustrate,  materials  already  provided.  However 
humble  this  employment,  he  ought  patiently  to  submit  to  it, 
whose  object  is  to  instruct,  not  those  who  have  made  consid- 
erable progress  in  the  mathematics,  but  those  who  are  just 
commencing  the  study.  Original  discoveries  are  not  for  the 
benefit  of  beginners,  though  they  may  be  of  great  importance 
to  the  advancement  of  science. 

The  arrangement  of  the  parts  is  such,  that  the  explanation 
of  one  is  not  made  to  dej^end  on  another  which  is  to  follow. 
The  addition,  multiplication,  and  division  of  powers,  for  in- 
stance, is  placed  after  involution.  In  the  statement  of  gen- 
eral rules,  if  they  are  reduced  to  a  small  nimiber,  their  ap- 
plications to  particular  cases  may  not,  always,  be  readily  un- 
derstood. On  the  other  hand,  if  they  are  very  numerous, 
they  become  tedious  and  burdensome  to  the  memory.  The 
rules  given  in  this  introduction,  are  most  of  them  compre- 
hensive ;  but  they  are  explained  and  applied,  in  subordinate 
articles. 

A  particular  demonstration  is  sometimes  substituted  for  a 
general  one,  when  the  application  of  the  principle  to  other 
cases  is  obvious.  The  examples  are  not  often  taken  from 
philosophical  subjects,  as  the  learner  is  supposed  to  be  fa- 
miliar with  none  of  the  sciences  except  arithmetic.  In  treat- 
ing of  negative  quantities,  frequent  references  are  made  to 
mercantile  concerns,  to  debt,  and  credit,  &c.  These  are 
merely  for  the  purpose  of  illustration.  The  whole  doctrine 
of  negatives  is  made  to  depend  on  the  single  principle,  that 


PREFACE.  vii 

they  are  quantities  to  be  subtracted.  But  the  student,  at 
this  early  period,  is  not  accustomed  to  abstraction.  He  re- 
quires particular  examples,  to  catch  his  attention,  and  aid  his 
conceptions. 

The  section  on  proportion,  will,  perhaps,  be  thought  use- 
less to  those  who  read  the  iiftli  Book  of  Euchd.  That  is  suf- 
ficient for  the  purposes  of  pure  geometrical  demonstration.  But 
it  is  important  that  the  propositions  should  also  be  presented 
under  the  algebraic  forms.  In  addition  to  this,  great  assis- 
tance may  be  derived  from  the  algebraic  notation,  in  demon- 
strating, and  reducmg  to  system,  the  laws  of  proportion.  The 
subject  instead  of  being  broken  up  into  a  multitude  of  dis- 
tinct propositions,  may  be  comprehended  in  a  few  general 
principles. 


CONTENTS, 


Page. 

Introductory  Observations,  on  the  Mathematics  in  general,     -  •  -  1 

ALGEBRA. 

Section  I.  Notation,  Positive  and  Negative  Quantities,  Axioms,  &c.         -  -      8 

II.  Addition, 21 

III.  Subtraction,     --------27 

IV.  MultipUcalion,        -  -  --  .  •  -81 
V.  Division,          -            -            -            -            -             -            *  -41 

VI.  Algebraic  Fractions,  ------  49 

VII.  Reduction  of  Equations,  by  Transposition,  Multiplication  and  Division, 

Solution  of  Problems,  -  -  -  -  -  -     65 

VIII.  Involution.     Notation,  Addition,  Subtract-on,  Multiplication,  and  Di- 
vision of  Powers,  ------  83 

IX.  Evolution.    Notation,  Reduction,  Addition,  Subtraction, Multiplication, 

Division,  and  Involution  of  Radical  Quantities,        -  -  -    99 

X.  Reduction   of  Equations   by  Involution  and   Evolution.       AfTected 

Quadratic  Equations,      ------         128 

XI.  Solution  of  Problems  which  contain  two  or  more  Unknown  Q,uantities,  153 
XII.  Ratio  and  Proportion,  -  -  -  -  -  -173 

XIII.  Variation  or  General  Proportion,  -  -  -  -         206 

XTV.  Arithmetical  and  Geometrical  Progression,      -  -  -  -  213 

XV.  Mathematical  Infinity,  ------        226 

XVI.  Division  by  Comi^ound  Divisors,  -----  233 

XVII.  Involution  of  Compound  duantities,  by  the  Binomial  Theorem,       -         242 
XVIII.  Evolution  of  Compound  duantitics,     -----  253 

XIX.  Infinite  Series,         ------  259 

XX.  Composition  and  Resolution  of  the  higher  Equations,  -  -  2*8 

XXI.  AppUcation  of  Algebra  to  Geometry,  -  -  «  -         292 

XXII.  Equations  of  Curves,  -  -.--.-308 


INTRODUCTORY  OBSERVATIONS 


ON    THE 


MATHEMATICS  IN  GENERAL. 


»<Ho« 


Art.  1.    Mathematics  is  the  science  of  quantity. 

Any  thing  which  can  be  multiplied,  divided,  or  measured,  is 
called  quantity.  Thus,  a  line  is  a  quantity,  because  it  can 
be  doubled,  trebled,  or  halved ;  and  can  be  measured,  by 
applying  to  it  another  line,  as  a  foot,  a  yard,  or  an  ell. 
Weight  is  a  quantit}',  which  can  be  measured,  in  pounds, 
ounces,  and  grains.  Time  is  a  species  of  quantity,  whose 
measure  can  be  expressed,  in  hours,  minutes,  and  seconds. 
But  color  is  not  a  quantity.  It  cannot  be  said,  with  propri- 
ety, that  one  color  is  twice  as  great,  or  half  as  great,  as 
another.  The  operations  of  the  mind,  such  as  thought, 
choice,  desire,  hatred,  &c.  are  not  quantities.  They  are  in- 
capable of  mensuration.* 

2.  Those  parts  of  the  Mathematics,  on  which  all  the 
others  are  founded,  are  Arithmetic,  Algebra,  and  Geometry. 

3.  Arith:metic  is  the  science  of  numbers.  Its  aid  is 
required  to  complete  and  apply  the  calculations,  in  almost 
every  other  department  of  the  mathematics. 

4.  Algebra  is  a  method  of  computing  by  letters  and  other 
sjanbols.  Fluxions,  or  the  Diflerential  and  Integral  Cal- 
culus, may  be  considered  as  belonging  to  the  higher  branches 
of  algebra,  f 

5.  Geometry  is  that  part  of  the  mathematics,  which  treats 
of  magnitude.  By  magnitude,  in  the  appropriate  sense  of 
the  term,  is  meant  that  species  of  quantit}^,  which  is  extend- 
ed; that  is,  which  has  one  or  more  of  the  three  dimensions, 
length,  breadth,  and  thickness.  Thus  a  line  is  a  magnitude, 
because  it  is  extended,  in  length.  A  swface  is  a  magnitude, 
having  length  and  breadth.     A  solid  is  a  magnitude,  having 

♦  See  Note  A.  t  See  Note  B. 


2  MATHEMATICS. 

length,  breadth,  and  thickness.  But  motion,  though  a  quan- 
tity, is  not,  strictly  speaking,  a.  magnitude.  It  has  neither 
length,  breadth,  nor  thickness.* 

6.  Trigonometry  and  Conic  Sections  are  branches  of 
the  mathematics,  in  which  the  principles  of  geometry  are 
applied  to  triangles,  and  the  sections  of  a  cone. 

7.  Mathematics  are  either  pure  or  mixed.  In  pure  mathe- 
matics, quantities  are  considered,  independently  of  any  sub- 
stances actually  existing.  But,  in  mixed  mathematics,  the 
relations  of  quantities  are  investigated,  in  connection  with 
some  of  the  properties  of  matter,  or  with  reference  to  the 
common  transactions  of  business.  Thus,  in  Surveying, 
mathematical  principles  are  applied  to  the  measuring  of 
land  ;  in  Optics,  to  the  properties  of  light ;  and  in  Astrono- 
my, to  the  motions  of  the  heavenly  bodies. 

8.  The  science  of  the  pure  mathematics  has  long  been 
distinguished,  for  the  clearness  and  distinctness  of  its  princi- 
ples ;  and  the  irresistible  conviction,  w^hich  they  cany  to  the 
mind  of  every  one  who  is  once  made  acquainted  with  them. 
This  is  to  be  ascribed,  partly  to  the  nature  of  the  subjects, 
and  partly  to  the  exactness  of  the  definitions,  the  axioms, 
and  the  demonstrations. 

9.  The  foundation  of  all  mathematical  knowledge  must 
be  laid  in  definitions.  A  definition  is  an  explanation  of  what 
is  meant,  by  any  word  or  phrase.  Thus,  an  equilateral  tri- 
angle is  defined,  by  saying,  that  it  is  a  figure  bounded  by 
three  equal  sides. 

It  is  essential  to  a  complete  definition,  that  it  perfectl)^  dis- 
tinguish the  thing  defined,  from  every  thing  else.  On  many 
subjects  it  is  difl^cult  to  give  such  precision  to  language,  that 
it  shall  convey,  to  every  hearer  or  reader,  exactly  tlie  same 
ideas.  But,  in  the  mathematics,  the  principal  terms  may  be 
so  defined,  as  not  to  leave  room  for  the  least  difference  of 
apprehension,  respecting  their  meaning.  All  must  be  agreed, 
as  to  the  nature  of  a  circle,  a  square,  and  a  triangle,  when 
they  have  once  learned  the  definitions  of  these  figures. 

Under  the  head  of  definitions,  may  be  included  explana- 
tions of  the  characters  which  are  used  to  denote  tlie  relations 
of  quantities.  Thus  the  character  V  i^  explained  or  defined, 
by  saying  that  it  signifies  the  same  as  the  words  square  root. 

10.  The  next  step,  after  becoming  acquainted  with  the 
meaning  of  mathematical  terms,  is  to  bring  them  together,  in 

♦  Some  writers,  however,  use  magnitude  as  synonymous  with  quantity. 


MATHEMATICS.  S 

the  form  of  propositions.  Some  of  the  relations  of  quantities 
require  no  process  of  reasoning,  to  render  them  evident.  To 
be  understood,  they  need  only  to  be  proposed.  That  a 
square  is  a  different  iigure  from  a  circle ;  that  the  whole  of  a 
thing  is  greater  than  one  of  its  parts ;  and  that  two  straight 
lines  cannot  enclose  a  space,  are  propositions  so  manifestly 
true,  that  no  reasoning  upon  them  could  make  them  more 
certain.  They  are,  therefore,  called  self-evident  truths,  or 
axioms. 

11.  There  are,  however,  comparatively  few  mathematical 
truths  w^hich  are  self-evident.  Most  require  to  be  proved  by 
a  chain  of  reasoning.  Propositions  of  this  nature  are  denom- 
inated theorems ;  and  the  process,  by  which  they  are  shown 
to  be  true,  is  called  demonstration.  This  is  a  mode  of  argu- 
ing, in  which,  every  inference  is  immediately  derived,  either 
from  definitions,  or  from  principles  which  have  been  previ- 
ously demonstrated.  In  this  way,  complete  certainty  is  made 
to  accompany  every  step,  in  a  long  course  of  reasoning. 

12.  Demonstration  is  either  direct  or  indirect.  The  for- 
mer is  the  common,  obvious  mode  of  conducting  a  demon- 
strative argument.  But  in  some  instances,  it  is  necessary  to 
resort  to  indirect  demonstration ;  which  is  a  method  of  es- 
tablishing a  proposition,  by  proving  that  to  suppose  it  not 
true,  would  lead  to  an  absurdity.  This  is  frequently  called 
reductio  ad  absurdum.  Thus,  in  certain  cases  in  geometry, 
two  lines  may  be  proved  to  be  equal,  by  shoAving  that  to  sup- 
pose them  unequal,  would  involve  an  absurdity. 

13.  Besides  the  principal  theorems  in  the  mathematics, 
there  are  also  Lemmas  and  Corollaries.  A  Lemma  is  a  pro- 
position which  is  demonstrated,  for  the  purpose  of  using  it,  in 
the  demonstration  of  soiue  other  proposition.  •  This  prepara- 
tory step  is  taken  to  prevent  the  proof  of  the  principal  theo- 
rem from  becoming  complicated  and  tedious. 

14.  A  Corollary  is  an  inference  from  a  preceding  proposi- 
tion. A  Scholium  is  a  remark  of  any  kind,  suggested. by 
something  which  has  gone  before,  though  not,  like  a  corolla- 
ry, iimnediately  depending  on  it. 

15.  The  immediate  object  of  inquiry,  in  the  mathematics, 
IS,  frequently,  not  the  demonstration  of  a  general  truth,  but 
a  method  of  performing  some  operation,  such  as  reducing  a 
vulgar  fraction  to  a  decimal,  extracting  the  cube  root,  or 
inscribing  a  circle  in  a  square.  This  is  called  solving  a  prob- 
lem. A  theorem  is  something  to  be  proved.  A  problem  is 
something  to  be  done. 


4  MATHEMATICS. 

16.  When  that  which  is  required  to  be  done,  is  so  easy,  as 
to  be  obvious  to  every  one,  without  an  explanation,  it  is  call- 
ed a  postulate.  Of  this  nature  is  the  drawing  of  a  straight 
line,  from  one  point  to  another. 

17.  A  quantity  is  said  to  be  given,  when  it  is  either  sup- 
posed to  be  already  known,  or  is  made  a  condition,  in  the 
statement  of  any  theorem  or  problem.  In  the  rule  of  pro- 
portion in  arithmetic,  for  instance,  three  terms  must  be  given 
to  enable  us  to  find  a  fourth.  These  three  terms  are  the 
data,  upon  which  the  calculation  is  foimded.  If  we  are  re- 
quired to  find  the  number  of  acres,  in  a  circular  island  ten 
miles  in  circumference,  the  circular  figure,  and  the  length  of 
the  circumference  are  the  data.  They  are  said  to  be  given 
by  supposition,  that  is,  by  the  conditions  of  the  problem.  A 
quantity  is  also  said  to  be  given,  when  it  may  be  directly  and 
easily  inferred  from  something  else  which  is  given.  Thus,  if 
two  numbers  are  given,  their  sum  is  given;  because  it  is  ob- 
tained, by  merely  adding  the  numbers  together. 

In  Geometry,  a  quantity  may  be  given,  either  in  position, 
or  magnitude,  or  both.  A  line  is  given  in  position,  when  its 
situation  and  direction  are  known.  It  is  given  in  magnitude, 
when  its  length  is  known.  A  circle  is  given  in  position,  when 
the  place  of  its  centre  is  known.  It  is  given  in  magnitude, 
when  the  length  of  its  diameter  is  known. 

18.  One  proposition  is  contrary,  or  contradictory  to  another, 
when,  Avhat  is  affirmed,  in  the  one,  is  denied,  in  the  other. 
A  proposition  and  its  contrary,  can  never  both  be  true.  It 
cannot  be  true,  that  two  given  lines  are  equal,  and  that  they 
are  not  equal,  at  the  same  time. 

19.  One  proposition  is  the  converse  of  another,  when  the 
order  is  inverted ;  so  that,  what  is  given  or  supposed  in  the 
first,  becomes  the  conclusion  in  the  last ;  and  what  is  given 
in  the  last,  is  the  conclusion,  in  the  first.  Thus,  it  can  be 
proved,  first,  that  if  the  sides  of  a  triangle  are  equal,  the'an- 
gles  are  equal ;  and  secondly,  that  if  the  angles  are  equal, 
the  sides  are  equal.  Here,  in  the  first  proposition,  the  equal- 
ity of  the  sides  is  given ;  and  the  equality  of  the  angles  in- 
ferred :  in  the  second,  the  equality  of  the  angles  is  given,  and 
the  equality  of  the  sides  inferred.  In  many  instances,  a  pro- 
position and  its  converse  are  both  true;  as  in  the  preceding 
example.  But  this  is  not  always  the  case.  A  circle  is  a 
figure  bounded  by  a  curve  ;  but  a  figure  bounded  by  a  curve 
is  not  of  course  a  circle. 


MATHEMATICS.  5 

20.  The  practical  applications  of  the  mathematics,  in  the 
common  concerns  of  business,  in  the  useful  arts,  and  in  the 
various  branches  of  physical  science  are  almost  inmunerable. 
Mathematical  principles  are  necessary  in  Mercantile  transac- 
tions, for  keeping,  arranging,  and  settling  accounts,  adjusting 
the  prices  of  commodities,  and  calculating  the  profits  of  trade  : 
in  J^avigation,  for  directing  the  course  of  a  ship  on  the  ocean, 
adapting  the  position  of  her  sails  to  the  direction  of  the  wind, 
finding  her  latitude  and  longitude,  and  determining  the  bear- 
ings and  distances  of  objects  on  shore  :  in  Surveying,  for 
measuring,  dividing,  and  laying  out  grounds,  taking  the  eleva- 
tion of  hilLs,  and  fixing  the  boundaries  of  fields,  estates,  and 
public  territories  :  in  Civil  Engineering,  for  constructing 
bridges,  aqueducts,  locks,  &c. :  in  Mechanics,  for  understand- 
ing the  laws  of  motion,  the  composition  of  forces,  the  equili- 
brium of  the  mechanical  powers,  and  the  structure  of  ma- 
chines :  in  ^.Architecture,  for  calculating  the  comparative 
strength  of  timbers,  the  pressure  which  each  will  be  required 
to  sustain,  the  forms  of  arches,  the  proportions  of  columns,  &c. : 
in  Fortification,  for  adjusting  the  position,  lines,  and  an- 
gles, of  the  several  parts  of  the  works  :  in  Gunneanj,  for  regu- 
lating the  elevation  of  the  cannon,  the  force  of  the  powder, 
and  the  velocity  and  range  of  the  shot :  in  Optics,  for  tracing 
tlie  direction  of  the  rays  of  light,  understanding  the  forma- 
tion of  images,  the  laws  of  vision,  the  separation  of  colors,  the 
nature  of  the  rainbow,  and  the  construction  of  microscopes 
and  telescopes :  in  Astronomy,  for  computing  the  distances, 
magnitudes,  and  revolutions  of  the  heavenly  bodies  ;  and  the 
influence  of  the  law  of  gravitation,  in  raising  the  tides,  dis- 
turbing the  motions  of  the  moon,  causing  the  return  of  the 
comets,  and  retaining  the  planets  in  their  orbits  :  in  Geogra- 
phy, for  determining  the  figure  and  dimensions  of  the  earth, 
the  extent  of  oceans,  islands,  continents,  and  countries  ;  the 
latitude  and  longitude  of  places,  the  courses  of  rivers,  the 
height  of  mountains,  and  the  boundaries  of  kingdoms :  in  His- 
tory, for  fixing  the  chronology  of  remarkable  events,  and 
estimating  the  strength  of  armies,  the  wealth  of  nations,  the 
value  of  their  revenues,  and  the  amount  of  their  population  : 
and,  in  the  concerns  of  Government,  for  apportioning  taxes, 
arranging  schemes  of  finance,  and  regulating  national  ex- 
penses. The  mathematics  have  also  important  applications 
to  Chemistry,  Mineralogy,  Music,  Painting,  Sculpture,  and 
indeed  to  a  great  proportion  of  the  whole  circle  of  arts  and 
sciences. 


6  MATHEMATICS. 

♦  21.  It  is  true,  that,  in  many  of  the  branches  which  have 
been  mentioned,  the  ordinary  business  is  frequently  trans- 
acted, and  the  mechanical  operations  performed,  by  persons 
who  have  not  been  regularly  instructed  in  a  course  of  mathe- 
matics. Machines  are  framed,  lands  are  surveyed,  and  ships 
are  steered,  by  men  who  have  never  thoroughly  investigated 
the  principles,  which  lie  at  the  foundation  of  their  respective 
arts.  The  reason  of  this  is,  that  the  methods  of  proceeding, 
in  their  several  occupations,  have  been  pointed  out  to  them, 
by  the  genius  and  labor  of  others.  The  mechanic  often 
works  by  rules,  which  men  of  science  have  provided  for  his 
use,  and  of  which  he  knows  nothing  more,  than  the  practical 
application.  The  mai-iner  calculates  his  longitude  by  tables, 
for  which  he  is  indebted  to  mathematicians  and  astronomers 
of  no  ordinary  attainments.  In  this  manner,  even  the  ab- 
struse parts  of  the  mathematics  are  made  to  contribute  their 
aid  to  the  common  arts  of  life. 

22.  But  an  additional  and  more  important  advantage,  to 
persons  of  liberal  education,  is  to  be  found,  in  the  enlarge- 
meiit  and  improvement  of  the  reasoning  powers.  The  mind, 
like  the  body,  acquires  strength  by  exertion.  The  art  of 
reasoning,  like  other  arts,  is  learned  by  practice.  It  is  per- 
fected, only  by  long  continued  exercise.  Mathematical  stu- 
dies are  peculiarly  fitted  for  this  discipline  of  the  mind. 
They  are  calculated  to  form  it  to  habits  of  fixed  attention ; 
of  sagacity,  in  detecting  sophistry  ;  of  caution,  in  the  admis- 
sion of  proof ;  of  dexterity,  in  the  arrangement  of  arguments ; 
and  of  skill,  in  making  all  the  parts  of  a  long  continued  pro- 
cess tend  to  a  result,  in  which  the  truth  is  clearly  and  firmly 
established.  When  a  habit  of  close  and  accurate  thinking 
is  thus  acquired,  it  may  be  applied  to  any  subject,  on  which 
a  man  of  letters  or  of  business  may  be  called  to  employ  his 
talents.  "  The  )^outh,"  says  Plato,  "  who  are  furnished  with 
mathematical  knowledge,  are  prompt  and  quick,  at  all  other 
sciences." 

It  is  not  pretended,  that  an  attention  to  other  objects  of 
inquiry  is  rendered  unnecessary,  by  the  study  of  the  mathe- 
matics. It  is  not  their  office,  to  lay  before  us  historical  facts  ; 
to  teach  the  principles  of  morals ;  to  store  the  fancy  with 
brilliant  images ;  or  to  enable  us  to  speak  and  write  with 
rhetorical  vigor  and  elegance.  The  beneficial  effects  which 
they  produce  on  the  mind,  are  to  be  seen,  principally,  in  the 
regulation  and  mcreased  energy  of  the  reasoning  powers 
These  they  are  calculated  to  call  into  frequent  and  vigorous 


MATHEMATICS.  7 

exercise.  At  the  same  time,  mathematical  studies  may  be 
so  conducted,  as  not  often  to  require  excessive  exertion  and 
fatigue.  Beginning  Avith  the  more  simple  subjects,  and  as- 
cending gradually  to  those  which  are  more  complicated,  the 
mind  acquires  strength  as  it  advances;  and  by  a  succession 
of  steps,  rising  regularly  one  above  another,  is  enabled  to 
surmount  the  obstacles  which  lie  in  its  way.  In  a  course  of 
mathematics,  the  parts  succeed  each  other  in  such  a  con- 
nected series,  that  the  preceding  propositions  are  preparatory 
to  those  which  follow.  The  student  who  has  made  himself 
master  of  the  former,  is  qualified  for  a  successful  investiga- 
tion of  the  latter.  But  he  who  has  passed  over  any  of  the 
ground  superficially,  will  find  that  the  obstructions  to  his 
future  progress  are  yet  to  be  removed.  In  mathematics  as  in 
war,  it  shovdd  be  made  a  principle,  not  to  advance,  while  any 
thing  is  left  unconquered  behind.  It  is  important  that  the 
student  should  be  cleqoly  impressed  with  a  conviction  of  the 
necessity  of  this.  Neither  is  it  sufficient  that  he  understands 
the  nature  of  one  proposition  or  method  of  operation,  before 
proceeding  to  another.  He  ought  also  to  make  himself  /a- 
miliar  with  every  step,  by  careful  attention  to  the  examples. 
He  must  not  expect  to  become  thoroughly  versed  in  the  sci- 
ence, by  merely  reading  the  main  principles,  rules,  and  obser- 
vations. It  is  practice  only,  which  can  put  these  completely 
in  his  possession.  The  method  of  studying  here  recom- 
mended, is  not  only  that  which  promises  success,  but  that 
which  will  be  found,  in  the  end,  to  be  the  most  expeditious, 
and  by  far  most  pleasant.  While  a  superficial  attention  oc- 
casions perplexity  and  consequent  aversion;  a  thorough 
investigation  is  rewarded  with  a  high  degree  of  gratification. 
The  pecuHar  entertainment  which  mathematical  studies  are 
calculated  to  furnish  to  the  mind,  is  reserved  for  those  who 
make  themselves  masters  of  the  subjects  to  which  their 
attention  is  called. 


Note. — The  principal  definitions,  theorems,  rules,  &c.  which  it  is  necessary 
to  commit  to  memory,  are  distinguished  by  being  put  in  Italics  or  Capitals. 


ALGEBRA. 

SECTION  I. 

NOTATION,  NEGATIVE  aUANTITIES,  AXIOMS,  &c. 

Art.  23.  ALGEBRA  may  be  defined,  a  general  method 
OF  investigating  the  relations  of  quantities,  by  let- 
ters, AND  other  symbols.  Tliis,  it  must  be  acknowledged, 
is  an  imperfect  account  of  the  subject;  as  every  account 
must  necessarily  be,  which  is  comprised  in  the  compass  of  a 
definition.  Its  real  nature  is  to  be  learned,  rather  by  an 
attentive  examination  of  its  parts,  than  from  any  summary 
description. 

The  solutions  in  Algebra,  are  of  a  more  general  nature 
than  those  in  common  Arithmetic.  The  latter  relate  to  par- 
ticular numbers ;  the  former  to  whole  classes  of  quantities. 
On  this  account.  Algebra  has  been  termed  a  kind  of  universal 
Arithmetic.  The  generality  of  its  solutions  is  principally 
owing  to  the  use  of  letters,  instead  of  numeral  figures,  to 
express  the  several  quantities  which  are  subjected  to  calcula- 
tion. In  Arithmetic,  when  a  problem  is  solved,  tlie  answer 
is  limited  to  the  particular  numbers  which  are  specified,  in 
the  statement  of  the  question.  But  an  Algebraic  solution 
may  be  equally  applicable  to  all  other  quantities  which  have 
the  same  relations.  This  important  advantage  is  owing  to 
the  difference  between  the  customary  use  of  figures,  and  the 
maimer  in  which  letters  are  employed  in  Algebra.  One  of 
the  nine  digits,  invariably  expresses  the  same  number :  but  a 
letter  may  be  put  for  any  number  whatever.  The  figure  8 
always  signifies  eight ;  the  figure  5,  five,  &c.  And,  though 
one  of  the  digits,  in  connection  with  others,  may  have  a  local 
value,  different  from  its  simple  value  when  alone ;  yet  the 
same  combination  always  expresses  the  same  number.  Tlius 
263  has  one  uniform  signification.  And  this  is  the  case  Avith 
every  other  combination  of  figures.  But  in  Algebra,  a  letter 
may  stand  for  any  quantity  which  we  wish  it  to  represent. 
Thus  6  may  be  put  for  2,  or  10,  or  50,  or  1000.  It  must  no^ 
be  understood  from  this,  however,  that  the  letter  has  no  de 


NOTATION.  9 

terminate  value.  Its  value  is  fixed  for  the  occasion.  For 
the  present  purpose,  it  remains  unaltered.  But  on  a  different 
occasion,  the  same  letter  may  be  put  for  any  other  number. 
A  calculation  may  be  greatly  abridged  by  the  use  of  let- 
ters ;  especially  when  very  large  numbers  are  concerned. 
And  when  several  such  numbers  are  to  be  combined,  as  in 
multiplication,  the  process  becomes  extremely  tedious.  But 
a  single  letter  may  be  put  for  a  large  number,  as  well  as 
for  a  small  one.  The  numbers  26347297,  68347823,  and 
27462498,  for  instance,  may  be  expressed  by  the  letters,  b,  c, 
and  d.  The  multiplying  them  together,  as  will  be  seen 
hereafter,  will  be  nothing  more  than  writing  them,  one  after 
another,  in  the  form  of  a  word,  and  the  product  will  be  sim- 
ply bed.  Thus  in  Algebra,  much  of  the  labor  of  calcula- 
tion may  be  saved,  by  the  rapidity  of  the  operations.  Solu- 
tions are  sometimes  effected,  in  the  compass  of  a  few  lines, 
which,  in  common  Arithmetic,  must  be  extended  through 
many  pages. 

24.  Another  advantage  obtained  from  the  notation  by  let- 
ters instead  of  figures,  is,  that  the  several  quantities  which 
are  brought  into  calculation,  may  be  preserved  distinct  from 
each  other  ;  though  carried  through  a  number  of  complicated 
processes ;  whereas,  in  arithmetic,  they  are  so  blended  to- 
gether, that  no  trace  is  left  of  what  they  were,  before  the 
operation  began. 

25.  Algebra  differs  farther  from  arithmetic,  in  making  use 
of  unknown  quantities,  in  carrying  on  its  operations.  In 
arithmetic,  all  the  quantities  Avhich  enter  into  a  calculation 
must  be  known.  For  they  are  expressed  in  numbers.  And 
every  number  must  necessarily  be  a  determinate  quantity. 
But  in  Algebra,  a  letter  may  be  put  for  a  quantity,  before 
its  value  has  been  ascertained.  And  yet  it  may  have  such 
relations  to  other  quantities,  with  which  it  is  connected,  as 
to  answer  an  UTiportant  purpose  in  the  calculation. 

NOTATION. 

26.  To  facilitate  the  investigations  in  algebra,  the  several 
steps  of  the  reasoning,  instead  of  being  expressed  in  words, 
are  translated  into  the  language  of  signs  and  symbols,  which 
may  be  considered  as  a  species  of  short-hand.  This  serves 
to  place  the  quantities  and  their  relations  distinctly  before 
the  eye,  and  to  bring  them  all  into  view  at  once.  They  are 
thus  more  readily  compared  and  understood,  than  when  rc- 

2 


10  ALGEBRA. 

moved  at  a  distance  from  each  other,  as  in  the  common 
mode  of  writing.  But  before  any  one  can  avail  himself  of 
this  advantage,  he  must  become  perfectly  familiar  with  the 
new  language. 

27.  The  quantities  in  algebra,  as  has  been  already  ob- 
served, are  generally  expressed  by  letters.  The  first  letters  of 
the  Alphabet  are  used  to  represent  known  quantities ;  and 
the  last  letters,  those  which  are  unknown.  Sometimes  the 
quantities,  instead  of  being  expressed  by  letters,  are  set  down 
in  figures,  as  in  common  arithmetic. 

28.  Besides  the  letters  and  figures,  there  are  certain  char- 
acters used,  to  indicate  the  relations  of  the  quantities,  or  the 
operations  which  are  performed  with  them.  Among  these 
are  the  signs  -\-  and  — ,  Vv^hich  are  read  plus  and  minus,  or 
more  and  less.  The  former  is  prefixed  to  quantities  which 
are  to  be  added;  the  latter,  to  those  which  are  to  be  sub- 
tracted. Thus  a-\-h  signifies  that  b  is  to  be  added  to  a.  It 
is  read  a  plus  b,  or  a  added  to  b,  or  a  and  b.  If  tlie  expres- 
sion be  a-bj  i,  e.  a  minus  b;  it  indicates  that  b  is  to  be  sub- 
tracted from  a. 

29.  The  sign  -f-  is  prefixed  to  quantities  which  are  con- 
sidered as  affirmative  or  positive ;  and  the  sign  — ,  to  t hose 
which  are  supposed  to  be  negative.  For  the  nature  of  this 
distinction,  see  art.  54. 

All  the  quantities  which  enter  into  an  algebraic  process, 
are  considered,  for  the  purposes  of  calculation,  as  either  posi- 
tive or  neg-\tive.  Before  the  first  one,  unless  it  be  negative,' 
the  sign  is  generally  omitted.  But  it  is  always  to  be  under- 
stood.    Thus  a-\-b,  is  the  same  as  -\-a-\-b. 

30.  Sometimes  both  -f-  ^md  —  are  prefixed  to  the  same 
letter.  The  sign  is  then  said  to  be  ambiguous.  Tbus  a^b 
signifies  that  in  certain  cases,  comprehended  in  a  general  so- 
lution, b  is  to  be  added  to  a,  and  in  other  cases  subtracted 
from  it. 

31.  When  it  is  intended  to  express  the  ditferenre  betAveen 
tAVO  quantities  without  deciding  which  is  the  one  to  be  sub- 
tracted, the  character  c/^  or  ^  is  used.  Thus  a^b,  or  acrb 
denotes  the  difference  between  a  and  b,  without  determining 
whether  a  is  to  be  subtracted  from  b,  or  b  from  a. 

32.  The  equality  between  two  quantities  or  sets  of  quanti- 
ties is  expressed  by  parallel  lines  =.  Thus  a-\-b  =  d  sig- 
nifies that  a  and  b  together  are  equal  to  c?.     And  a-{-</  =  c 


NOTA.TION.  11 

=  b-\-g=h  signifies  that  a  and  d  equal  c,  which  is  equal  to 
b  and  g,  which  are  equal  to  h.  So  8-f  4=16  -4  =  10-1-2  = 
74.2-^3  =  12. 

33.  When  the  first  of  the  two  quantities  compared,  is 
greater  than  the  other,  the  character^  is  placed  between 
them.     Thus  a^b  signifies  that  a  is  greater  than  b. 

If  the  first  is  less  than  the  other,  the  character  <^  is  used ; 
as  a<Cb ;  i.  e.  a  is  less  than  b.  In  both  cases,  the  quantity 
towards  which  the  character  opens,  is  greater  than  the  other. 

34.  A  numeral  figure  is  often  prefixed  to  a  letter.  This 
is  called  a  co-efficient.  It  shows  how  often  the  quantity  ex- 
pressed by  the  letter  is  to  be  taken.  Thus  26  signifies  twice 
b  ;  and  96,  9  times  6,  or  9  nmltiplied  into  b. 

The  co-efficient  may  be  either  a  whole  number  or  a  frac- 
tion. Thus  §6  is  two-thirds  of  b.  When  the  co-efficient  is 
not  expressed,  1  is  always  to  be  understood.  Thus  a  is  the 
same  as  la;  i.  e.  once  a. 

35.  The  co-eflicient  may  be  a  letter,  as  well  as  a  figure. 
In  the  quantity  ?)i6,  m  may  be  considered  the  co-efl[icient  of 
b ;  because  b  is  to  be  taken  as  many  times  as  there  are  units 
in  m.  If  m  stands  for  6,  then  mb  is  6  times  b.  In  3a6c,  3 
may  be  considered  as  the  co-efficient  of  abc ;  Sa  the  co-effi- 
cient of  be  ;  or  Sab,  the  co-efficient  of  c.     See  art.  42. 

S6.  A  simple  quantity  is  either  a  single  letter  or  number, 
or  several  letters  connected  together  without  the  signs -j- 
and-.  Thus  a,  ab,  abd  and  Sb  are  each  of  them  simple 
quantities.  A  comppund  quantity  consists  of  a  number  of 
simple  quantities  connected  by  the  sign  -f-  or  - .  Thus  a-\- 
b,  d  —  y,  b  —  d-{-Sh,  are  each  compound  quantities.  The  mem- 
bers of  which  it  is  composed  are  called  terms. 

37.  If  there  are  two  terms  in  a  compound  quantity,  it  is 
called  a  binomial.  Thus  a-\-b  and  a-b  are  binomials.  The 
latter  is  also  called  a  residual  quantity,  because  it  expresses 
the  difference  of  two  quantities,  or  the  remainder,  after  one  is 
taken  from  the  other.  A  compound  quantity  consisting  of 
three  terms,  is  sometimes  called  a  tnnomial;  one  of  four  terms, 
a  quadrinomial,  &c. 

38.  When  the  several  members  of  a  compound  quantity 
are  to  be  subjected  to  the  same  operation,  they  are  frequent- 
ly connected  by  a  line  called  a  vinculum.  Thus  a  -  b-\-c 
shows  that  the  sum  of  b  and  c  is  to  be  subtracted  from  a.  But 
a  -  6-j-c  signifies  that    b  only  is  to   be    subtracted    from  a 


12  ALGEBRA. 

while  c  is  to  be  added.  The  sum  of  c  and  d,  subtracted 
from  the  sum  of  a  and  b,  is  a^b  -  c-\-d.  The  marks  used 
for  parentheses,  (  ),  are  often  substituted  instead  of  a  hne,  for 
a  vinculum.  Thus  x  -  {a-{-c)  is  the  same  as  a:  -  a-\-c.  The 
equality  of  two  sets  of  quantities  is  expressed,  without  using 
a  vinculum.  Thus  a-\-b=c-{-d  signifies,  not  that  b  is  equal 
to  c ;  but  that  the  sum  of  a  and  b  is  equal  to  the  sum  of  c 
and  d. 

39.  A  single  letter,  or  a  number  of  letters,  representing  any 
quantities  with  their  relations,  is  called  an  algebraic  expres- 
sion; and  sometimes  a  formula.  Thus  a-\-b-\-Sd  is  an 
algebraic  expression, 

40.  The  character  X  denotes  multiplication.  Thus  aX& 
is  a  multiplied  into  b:  and  6x3  is  6  times  3,  or  6  into  3. 
Sometimes  a  point  is  used  to  indicate  multiplication.  Thus 
a.  6  is  the  same  as  axb.  But  the  sign  of  multiplication  is 
more  commonly  omitted,  between  simple  quantities ;  and 
the  letters  are  connected  together,  in  the  form  of  a  word  or 
syllable.  Thus  ab  is  the  same  as  « .  6  or  a  X  ^-  And  bcde 
is  the  same  as  bXf^Xdx^-  When  a  compound  quantity  is 
to  be  multiplied,  a  vinculum  is  used,  as  in  the  case  of  sub- 
traction. Thus  the  sum  of  a  and  b  multiplied  into  the  sum 
of  c  and  d,  is  a+6  X  c-frf,  or  {a-\-b)  X  (c^d).  And 
(6-f2)  X  5  is  8  X  5  or  40.  But  6  +  2x5  is  6-|-10  or  16. 
When  the  marks  of  parentheses  are  used,  the  sign  of  multi- 
phcation  is  frequently  omitted.    Thus  {x-\-y)  (^  -  y)  is  (a?+2/) 

X  (x-y-) 

4L  When  two  or  more  quantities  are  multiplied  together, 
each  of  them  is  called  a  factor.  Jn  the  product  ab,  a  is  a 
factor,  and  so  is  b.  In  the  product  a:X«-fw,  x  is  one  of  the 
factors,  and  a-f-m,  the  other.  Hence  every  co-efjUcient  may  be 
considered  a  factor.  (Art.  35.)  In  the  product  Sy,  3  is  a 
factor  as  well  as  y. 

42.  A  quantity  is  said  to  be  resolved  into  factors,  when  any 
factors  are  taken,  which,  being  multiplied  together,  will  pro- 
duce the  given  quantity.  Thus  Sab  may  be  resolved  into 
the  two  factors  3a  and  b,  because  3ax^  is  Sab.  And  5amn 
may  be  resolved  into  the  three  factors  5a,  and  m,  and  n. 
And  48  may  be  resolved  into  the  two  factors  2  X  24,  or  3  X 16, 
<5r  4x12,  or  6x8 ;  or  into  the  three  factors  2x^x8,  or  4 x 
6X2,  &c. 


NOTATION.  13 

43.  The  character  -f-  is  used  to  show  that  the  quantity 
which  precedes  it,  is  to  be  divided,  by  that  which  follows. 
Thus  a-^c  is  a  divided  by  c :  and  a-\-b-^c-]-d  is  the  sum 
of  a  and  b,  chvided  by  the  sum  of  c  and  d.  But  in  algebra, 
division  is  more  commonly  expressed,  by  writing  the  divisor 
under  the  dividend,  in  the  form  of  a  vulgar  fraction.     Thus 

,  is  the  same  as  a-^b :  and  ,  ,  ,  is  the  difference  of  c  and  b 
b  d-{-h 

divided  by  the  sum  of  d  and  h.     A  character  prefixed  to  the 

dividing  line  of  a  fractional  expression,  is  to  be  understood 

as  referring  to  all  the  parts  taken  collectively  ;  that  is  to  the 

b-[-c 

whole  value  of  the  quotient.     Thus  a — -    signifies  that 

^  m-\-n     ° 

the  quotient  of  b-\-c  divided  by  m-\-n  is  to  be  subtracted  from  a. 

.     -    c-d  ^^  h4-n    ,         •      ,         ,      ^  .         . 

And  — ; —  X denotes  that  the  iirst  quotient  is  to  be 

a-{-m       x  —  y 

multiplied  into  the  second. 

44.  When  four  quantities  are  proportional,  the  proportion 
is  expressed  b}^  points,  in  the  same  manner,  as  in  the  Rule  of 
Three  in  arithmetic.  Thus  a:b::c:d  signifies  that  a  has  to 
6,  the  same  ratio  which  c  has  to  d.  And  ab:  cd:  :  a-\-m  : 
b-\-n,  means,  that  ab  is  to  cd  ;  as  the  sum  of  a  and  m,  to  the 
sum  of  b  and  n. 

45.  Algebraic  quantities  are  said  to  be  alike,  when  they 
are  expressed  by  the  same  letters,  and  are  of  the  same  power: 
and  unlike,  when  the  letters  are  different,  or  when  the  same 
letter  is  raised  to  different  powers.*  Thus  ab,  Sab,  -ab, 
and  -6ab,  are  like  cpiantities,  because  the  letters  are  the 
same  in  each,  although  the  signs  and  co-efficients  are  differ- 
ent. But  3a,  Sy,  and  Sbx,  are  unlike  quantities,  because 
the  letters  are  unlike,  although  there  is  no  difference  in  the 
signs  and  co-efficients. 

46.  One  quantity  is  said  to  be  a  multiple  of  another,  when 
the  former  contains  the  latter  a  certain  number  of  times  with- 
out a  remainder.  Thus  10a  is  a  multiple  of  2a;  and  24  is 
a  multiple  of  6. 

47.  One  quantity  is  said  to  be  a  measure  of  another,  when 
the  former  is  contained  in  the  latter,  any  number  of  times, 
without  a  remainder.  Thus  36  is  a  measufe  of  I5b;  and  7 
is  a  measure  of  35. 


♦  For  the  notation  of  powers  and  roots,  see  the  sections  on  those  subjects. 


14  ALGEBRA. 

48.  The  value  of  an  expression,  is  the  number  or  quantity, 
for  which  the  expression  stands.  Thus  the  value  of  3-|-4  is 
7;  of  3x4  is  12;  of  V' is  2. 

49.  The  RECIPROCAL  of  a  quantity,  is  the  quotient  urising 
from  dividing  a  unit  by  that  quantity.     Thus  the  reciprocal 

of  a  is  -  ;  the  reciprocal  of  a-{-b  is         ,  ;  the  reciprocal  of  4 

.    1 

50.  The  relations  of  quantities,  which  in  ordinarj^  language, 
are  signified  by  words,  are  represented  in  the  algebraic  nota- 
tion, by  signs.  The  latter  mode  of  expressing  these  rela- 
tions, ought  to  be  made  so  familiar  to  the  mathematical 
student,  that  he  can,  at  any  time,  substitute  the  one  for  the 
other.  A  few  examples  are  here  added,  in  which,  words 
are  to  be  converted  into  cigns. 

1.  What  is  the  algebraic  expression  for  the  following 
statement,  in  which  the  letters  a,  b,  C,  &c.  may  be  supposed 
to  represent  any  given  quantities  1 

The  product  of  a,  b,  and  c,  divided  by  the  difference  of  c 
and  d,  is  equal  to  the  sum  of  b  and  c  added  to  15  times  h. 

Ans.  — j  =  b^c-\-15h. 

2.  The  product  of  the  difference  of  a  and  h  into  the  sum 
of  b,  c,  and  d,  is  equal  to  37  times  m,  added  to  the  quotient 
of  b  divided  by  the  sum  of  h  and  b.  Ans. 

3.  The  sum  of  a  and  b,  is  to  the  quotient  of  b  divided  by 
c;  as  the  product  of  a  into  c,  to  12  times  h.  Ans. 

4.  The  sum  of  a,  b,  and  c,  divided  bj^  six  times  their  pro- 
duct, is  equal  to  four  times  their  smn  diminished  by  d.  Ans. 

5.  The  quotient  of  6  divided  by  the  sum  of  a  and  b,  is 
equal  to  7  times  d,  diminished  by  the  quotient  of  b,  divided 
by  36.  Ans. 

51.  It  is  necessary  also,  to  be  able  to  reverse  what  is  done 
in  the  preceding  examples,  that  is,  to  ^anslate  the  algebraic 
signs  into  common  language. 

What  Avill  the  following  expressions  become,  when  words 

are  substituted  for  the  signs  ? 

,    a-\-b       ,     '  „  a 

1.  —i — =abc-6m  ' 


h    "~  '   a-{-c 

Ans.  The  sum  of  a  and  b  divided  by  h,  is  equal  to  the 
product  of  a,  b,  and  c  diminished  by  6  times  m,  and  increased 
by  the  quotient  of  a  divided  by  the  siun  of  a  and  c. 


NOTATION.  15 


Sh-c 


2.  a6+!lll_l=:  d  X  a+6-f  c 


m 


x-^y  6+6 

3.  a+7  {h+x)-.ll^={a^h)  (b-c). 

4.  a-b:ac::  '—  :  3  x  M-^^-rV- 

m 

^       a-h  d-^ah  _jjayjri^^^^     ^d 

3-J-6-  c  2m  am  h~\-d 

52.  At  the  close  of  an  algebraic  process,  it  is  frequently 
necessary  to  restore  the  numbers^  for  which  letters  had  been 
substituted,  at  the  beginning-.  In  doing  this,  the  sigio  of  mul- 
tiplication must  not  be  omitted,  as  it  genejally  is,  between 
factors,  expressed  by  letters.  Thus,  if  a  stauds  for  3,  and  b 
for  4 ;  the  product  ah  is  not  34,  but  3x4,  i.  e.  12. 

In  the  following  examples, 

Let  a=S  And  d=6, 

b  =  4  mz=8. 


71=10. 


Then    1   «+^,^_c-n_3+8     4x2-10^ 
'     '     cd  '^    Sd       2x6         3x6 
^  ^+«^^-6cmn   •  d-4cn 


c  —  dm  Bab 

3.  bm  ^_^^b-Sd_Sbn-bc  ^    b_^ 


cdm        4a-\-Scd      a 

53.  An  algebraic  expression,  in  which  numbers  have  been 
substituted  for  letters,  may  often  be  rendered  much  more 
simf^e,  by  reducing  several  terms  to  one.  This  cannot 
generally  be  done,  while  the  letters  remain.  .  If  a-\-b  is  used 
for  the  sum  of  two  quantities,  a  cannot  be  united  in  the  same 
term  with  b.  But  if  a  stands  for  3,  and  b  for  4,  then  a-{-b 
=  ^+4  =  7.  The  value  of  an  expression,  consistinsr  of  many 
terms  may  thus  be  found,  by  actually  performing,  Avith  the 
numbers,  the  operations  of  addition,  subtraction,  multipUca- 
tion,  &c.  indicated  by  the  algebraic  characters. 

Find  the  value  of  the  following  expressions,  in  which  the 
letters  are  supposed'  to  stand  for  the  same  nmnbers,  as  in  the 
preceding  article. 

1.  lff+a+mn=^+34-8xl0  =  9+3+8O=92. 


16  ALGEBRA. 


2.  afcm+_:^+2n=3x4x8+?^+2xl0: 
m-b  8-6 

m-b 


3.  a-\-cXn-m-{- -ft  X  n-m= 

m-d 


4.°X'^+%afcc-£+L>< 


n  -  d  ?i  -  6 


n-o  c 


^ac+5m     ^^_  4ci-6xa-o^ 

POSITIVE  AND  NEGATIVE  QUANTITIES.* 

54.  To  one  who  has  just  entered  on  the  study  of  algebra, 
there  is  generally  nothing  moie  perplexing,  than  the  use  of 
what  are  called  negative  quantities.  He  supposes  he  is  about 
to  be  introduced  to  a  class  of  quantities  which  are  entirely 
new  ;  a  sort  of  mathematical  nothings,  of  which  he  can  form 
no  distinct  conception.  As  positive  quantities  are  real,  he 
concludes  that  those  which  are  negative  must  be  imaginary. 
But  this  is  owing  to  a  misapprehension  of  the  term  negative, 
as  used  in  the  mathematics. 

55.  A  NEGATIVE  quantity  is  one  which  is  required 
TO  BE  SUBTRACTED.  When  several  quantities  enter  into 
a  calculation,  it  is  frequently  necessary  that  some  of  them 
should  be  added  together,  while  others  are  subtracted.  The 
former  are  called  affirmative  or  positive,  and  are  marked  with 
the  sign  -\-  ;  the  latter  are  termed  negative,  and  distinguished 
by  the  sign  -.  If,  for  instance,  the  profits  of  trade  are  the 
subject  of  calculation,  and  the  gain  is  considered  positive ; 
the  loss  will  be  negative ;  because  the  latter  must  be  subtracted 
from  the  former,  to  determine  the  clear  profit.  If  the  sums 
of  a  book  account  are  brought  into  an  algebraic  process,  the 
debt  and  the  credit  are  distinguished  by  opposite  signs.  If  a 
man  on  a  journey  is,  by  any  accident,  necessitated  to  return 
several  miles,  this  backward  motion  is  to  be  considered  nega- 
tive, because  that,  in  determining  his  real  progress,  it  must 
be  subtracted  from  the  distance  which  he  has  travelled  in 
the  opposite  direction.  If  the  ascent  of  a  body  from  the  earth 
be  called  positive,  its  descent  will  be  negative.  These  are 
only  different  examples  of  the  same  general  principle.      In 

*  On  the  subject  of  negative  Quantities,  see  Newton's  Universal  Aritlinietic, 
Maseres  on  the  Negative  Sign,  Mansfield's  Mathematical  Essays,  and  Mac- 
laurin's,  Simpson's,  Euler's,  Saunderson's,  gind  Ludlam's  Algebra. 


NEGATIVES.  17 

each  of  the  mstances,  one  of  the  quantities  is  to  be  suhiracted 
from  the  other. 

6Q.  Tlie  terms  positive  and  negative,  as  used  in  the  mathe- 
matics, are  merel}^  relatke.  They  imply  that  there  is,  either 
in  the  nature  of  tlie  quantities,  or  in  their  circumstances,  or 
in  the  purposes  which  they  are  to  answer  in  calculation, 
some  sucii  opposition  as  requires  that  one  should  be  subtracted 
from  the  other.  But  this  opposition  is  not  that  of  existence  and 
non-existence,  nor  of  one  thing  greater  than  nothing,  and 
another  less  than  nothing.  For,  in  many  cases,  either  or 
the  signs  may  be,  indifferently  and  at  pleasure,  applied  to 
the  very  same  quantity ;  that  is,  the  two  characters  may 
change  places.  In  determining  the  progress  of  a  ship,  for 
instance,  her  easting  may  be  marked  -f-  j  and  her  westing  -  ; 
or  the  westing  may  be  -j- ,  and  the  easting  - .  All  that  is 
necessary  is,  that  the  two  signs  be  prefixed  to  the  quantities, 
in  such  a  manner  as  to  show,  which  are  to  be  added, 
and  which  subtracted.  In  different  processes,  they  maj 
be  differently  applied.  On  one  occasion,  a  doAvnward  mo- 
tion may  be  called  positive,  and  on  another  occasion  negative. 

57.  In  every  algebraic  calculation,  some  one  of  the  quan- 
tities must  be  fixed  upon,  to  be  considered  positive.  All 
other  quantities  which  will  increase  tliis,  must  be  positive  also. 
But  those  which  will  tend  to  diminish  it,  must  be  negative. 
In  a  mercantile  concern,  if  the  stock  is  supposed  to  be  positive, 
tl^e  profits  will  be  positive  ;  for  they  increase  the  stock  ;  they 
are  to  be  added  to  it.  But  the  losses  will  be  negative  ;  for 
they  diminish  the  stock ;  they  are  to  be  subtracted  from  it. 
When  a  boat,  in  attempting  to  ascend  a  river,  is  occasionally 
driven  back  by  the  current ;  if  the  progress  up  the  stream,  te 
any  particular  point,  is  considered  positive,  every  succeeding 
instance  of  forward  motion  will  be  positive,  while  the  back^ 
ward  motion  will  be  negative. 

58.  A  negative  quantity  is  frequently  greater,  than  the 
positive  one  with  which  it  is  comiected.  But  how,  it  may 
be  asked,  can  the  former  be  subtracted  from  the  latter  1  The 
greater  is  certainly  not  contained  in  the  less  :  how  then  can 
it  be  taken  out  of  it  ?  The  answer  to  this  is,  that  the  greater 
may  be  supposed  first  to  exhaust  the  less,  and  then  to  leave 
a  remainder  equal  to  the  difference  between  the  two.  If  a 
man  has  in  his  possession  1000  dollars,  and  has  contracted  a 
debt  of  1 500  ;  the  latter  subtracted  from  the  former,  not 
only  exhausts  the  whole  of  it,  but  leaves  a  balance  of  500 

3 


IS  ALGEBRA. 

against  him.     In  common  language,  he  is  500  dollars  worse 
than  nothing. 

59.  In  this  way,  it  frequently  happens,  in  the  course  of  an 
algebraic  process,  that  a  negative  quantity  is  brought  to  stand 
alone.  It  lias  the  sign  of  subtraction,  without  being  con- 
nected with  any  other  quantity,  from  which  it  is  to  be  sub- 
tracted. This  denotes  that  a  previous  subtraction  has  left  a 
remainder,  which  is  a  part  of  the  quantity  subtracted.  If 
the  latitude  of  a  ship  Avhich  is  20  degrees  north  of  the  equator, 
is  considered  positive,  and  if  she  sails  south  25  degrees  ;  her 
motion  fust  diminishes  her  latitude,  then  reduces  it  to  noth- 
ing, and  finally  gives  her  5  degrees  of  south  latitude.  The 
sign  -  prefixed  to  the  25  degrees,  is  retained  before  the  5, 
to  show  that  this  is  what  remains  of  the  southward  motion, 
after  balancing  the  20  degrees  of  north  latitude.  If  the  mo- 
tion southward  is  only  15  degrees,  the  remainder  nmst  be 
-[-5,  instead  of -5,  to  show  that  it  is  a  part  of  the  ship's 
northern  latitude,  which  has  been  thus  far  diminished,  but  not 
reduced  to  nothing.  The  balance  of  a  book  account  will  be 
positive  or  negative,  according  as  the  debt  or  the  credit  is  the 
greater  of  tl\e  two.  To  determine  to  whsch  side  the  remain- 
der belongs,  the  sign  must  be  retained.  Though  there  is  no 
other  quantity,  from  which  this  is  again  to  be  subtracted,  or  to 
which  it  is  to  be  added. 

60.  When  a  quantity  continually  decreasing  is  reduced  to 
nothing,  it  is  sometimes  said  to  become  afterwards  less  than 
nothing.  But  this  is  an  exceptionable  manner  of  speaking.* 
No  quantity  can  be  really  less  than  nothing.  It  may  be  di- 
minished, till  it  vanishes,  and  gives  place  to  an  opposite  quan- 
tity. The  latitude  of  a  ship  crossing  the  equator,  is  first 
made  less  than  nothing,  and  afterwards  contrary  to  what  it 
was  before.  The  north  and  south  latitudes  may  therefore 
be  properly  distinguished,  by  the  signs  -|-  and  -  ;  all  the 
positive  degrees  being  on  one  side  of  0,  and  all  the  negative, 
on  the  other  ;  thus, 

+6,  +5,  +4,  +3,  +2,4-1,  0,  -  1,  -  2,  -  3,  -  4,  -  5,  &c. 

The  numbers  belonging  to  any  other  series  of  opposite 
quantities,  may  be  arranged  in  a  similar  manner.  So  that 
0  may  be  conceived  to  be  a  kind  of  dividing  point  between 

*  The  expression  "/es,?  than  nofhins:"  may  not  be  M^holly  improper;  if  it  is 
intended  to  be  understood,  not  literally,  but  merely  as  a  convenient  pliraso 
adopted  for  the  sake  of  avoiding  a  tedious  circumlocution  ;  as  we  say  "the  sun 
rises,"  instead  of  saying  "the  earth  rolls  round,  and  brings  the  sun'into  view." 
The  use  of  it  in  tliis  manner,  is  warranted  by  Newton,  Euler  and  otJiers. 


AXIOMS.  19 

positive  and  negative  numbers.  On  a  thermometer,  the  de- 
grees above  0  may  be  considered  positive,  and  those  below  0, 
negative. 

61.  A  quantity  is  sometimes  said  to  be  subtracted  from  0. 
By  this  is  meant,  that  it  belongs  on  the  negative  side  of  0. 
But  a  quantit}^  is  said  to  be  added  to  0,  when  it  belongs  on 
the  positive  side.  Thus,  in  speaking  of  the  degrees  of  a 
thermometer,  O-j-6  means  6  degrees  above  0;  aiid  0-6,  6 
degrees  below  0. 

AXIOMS. 

62.  The  object  of  mathematical  inquiry  is,  generally,  to 
investigate  some  unknown  quantity,  and  discover  how  great 
it  is.  This  is  elTected,  by  comparing  it  Avith  some  other 
quantity  or  quantifies  already  known.  The  dimensions  of 
a  stick  of  timber,  are  found,  by  applying  to  it  a  measuring 
rule  of  known  length.  The  weight  of  a  body  is  ascertained, 
by  placing  it  in  one  scale  of  a  balance,  and  observing  how 
many  pounds  in  the  opposite  scale,  will  equal  it.  And  any 
quantity  is  determined,  when  it  is  found  to  be  equal  to  some 
known  quantity  or  quantities. 

Let  a  and  6  be  known  quantities,  and  y,  one  which  is  un- 
known.    Then  y  will  become  known,  if  it  be  discovered  to 
be  equal  to  the  sum  of  a  and  b  :  that  ig"  if 
y  =  a-\-b. 

An  expression  like  this,  representing  the  equality  between 
one  quantity  or  set  of  quantities,  and  another,  is  called  an 
equation.  It  will  be  seen  hereafter,  that  much  of  the  business 
of  algebra  consists  in  finding  equations,  in  which  some  un- 
known quantity  is  shown  to  be  equal  to  others  which  are 
known.  But  it  is  not  often  the  fact,  that  the  first  compari- 
son of  the  quantities,  furnishes  the  equation  required.  It 
will  generally  be  necessary  to  make  a  number  of  additions, 
subtractions,  multiplications,  &c.  before  the  imknown  quanti- 
ty is  discovered.  But  in  all  these  changes,  a  constant  equality 
must  be  preserved,  between  the  two  sets  of  quantities  com- 
pared. This  will  be  done,  if,  in  making  the  alterations,  we 
are  guided  by  the  following  axioms.  These  are  not  inserted 
here,  for  the  purpose  of  being  proved ;  for  they  are  self- 
evident.  (Art.  10.)  But  as  they  must  be  continually  intro- 
duced or  implied,  in  demonstrations  and  the  solutions  of 
problems,  thev  are  placed  together,  for  the  convenience  of 
reference 


20  ALGEBRA. 

63.  Axiom  1.  If  the  same  quantity  or  equal  quantities  be 
added  to  equal  quantities,  their  sums  will  be  equal. 

2.  If  the  same  quantity  or  equal  quantities  be  subtracted 
from  equal  quantities,  the  remainders  will  be  equal. 

3.  If  equal  quantities  be  multiplied  into  the  same,  or  equal 
quantities,  the  products  will  be  equal. 

4.  If  equal  quantities  be  divided  by  the  same  or  equal 
quantities,fthe  quotients  will  be  equal. 

^^    %  5.   If  the   same  quantity  be  both  added  to  and  subtracted 
from  another,  the  value  of  the  latter  will  not  be  altered. 

6.  If  a  quantity  be  both  midtiplied  and  divided  by  another, 
the  value  of  the  former  will  not  be  altered. 

7.  If  to  unequal  quantities,  equals  be  added,  the  greater 
will  give  the  greater  sum. 

8.  If  from  unequal  quantities,  equals  be  subtracted,  the 
greater  will  give  the  greater  remainder. 

9.  If  unequal  quantities  be  nmltiplied  b}^  equals,  the 
greater  will  give  the  greater  product. 

10.  If  unequal  quantities  be  divided  by  equals,  the  greater 
will  give  the  greater  quotient, 

11.  Quantities  which  are  respectively  equal  to  any  other 
quantity  are  equal  to  each  other. 

12.  The  whole  of  a  quantity  is  greater  than  a  part. 

This  is,  by  no  means,  a  complete  list  of  the  self-evident 
propositions,  which  are  furnished  by  the  mathematics.  It  is 
not  necessary  to  enumerate  them  all.  Those  have  been 
selected,  to  which  we  shall  have  the  most  frequent  occasion 
to  refer. 

64.  The  investigations  in  algebra  are  carried  on,  princi- 
pally, by  means  of  a  series  of  equations  and  jwoportions.  But 
instead  of  entering  directly  upon  these,  it  will  be  necessary 
to  attend  in  the  first  place,  to  a:  number  of  processes,  on 
which  the  management  of  equations  and  proportions  de- 
pends. These  preparatory  operations  are  similar  to  the  cal- 
culations under  the  common  rules  of  arithmetic.  We  have 
addition,  multiplication,  division,  involution,  &c.  in  algebra, 
as  well  as  in  arithmetic.  But  this  application  of  a  common 
name,  to  operations  in  these  two  branches  of  the  mathemat- 
ics, is  often  the  occasion  of  perplexity  and  mistake.  The 
learner  naturally  expects  to  find  addition  in  algebra  the  same 
as  addition  in  arithmetic.  They  are  in  fact  the  same,  in 
many  respects  :  in  all  respects  perhaps,  in  which  the  steps  of 
the  one  wdll  admit  of  a  direct  comparison,  with  those  of  the 
other      But  addition  in  algebra  is  more  extensive,  than  in 


ADDITION.  21 

arithmetic.  Tlie  same  observation  may  be  made  concerning 
several  other  operations  in  algebra.  They  are,  in  many 
points  of  vicAV,  the  same  as  those  which  bear  the  same  names 
in  arithmetic.  But  they  are  frequently  extended  farther,  and 
comprehend  processes  Avhich  are  miknown  to  arithmetic. 
This  is  commonly  owing  to  the  introduction  of  negative 
quantities.  The  management  of  these  requires  steps  which 
are  unnecessary,  where  quantities  of  one  class  only  are  con- 
cerned. It  will  be  important,  therefore,  as  we  pass  along,  to 
mark  the  difference  as  well  as  tlie  resemblance,  between  arith- 
metic and  algebra ;  and,  in  some  instances,  to  give  a  new 
definition,  accommodated  to  the  latter. 


SECTION  II. 


ADDITION. 

Art.  65.  In  entering  on  an  algebraic  calculation,  the  first 
thing  to  be  done,  is  evidently  to  collect  the  materials.  Seve- 
ral distinct  quantities  are  to  be  concerned  in  the  process. 
These  nmst  be  brought  together.  They  must  be  connected 
in  some  form  of  expression,  which  will  present  them  at  once 
to  our  \4ew,  and  sliow  the  relations  which  they  have  to  each 
other.  This  collecting  of  quantities  is  what,  in  algebra,  is 
called  ADDITION.      It  may  be  defined,  the  connecting  of 

SEVERAL  quantities,  WITH  THEIR  SIGNS,  IN  ONE  ALGEBRAIC 
EXPRESSION. 

66.  It  is  common  to  include  in  the  definition,  "uniting  in 
one  term,  such  quantities,  as  will  admit  of  being  united." 
But  this  is  not  so  much  a  part  of  the  addition  itself,  as  a 
reduction,  which  accompanies  or  follows  it.  The  addition 
may,  in  all  cases  be  performed,  by  merely  connecting  the 
quantities  by  their  proper  signs.  Thus  a  added  to  b,  is  evi- 
dently a  and  b  :  that  is,  according  to  the  algebraic  notation, 
a-\-b.  And  a  added  to  the  sum  of  b  and  c,  is  a-\-b-\-c.  And 
a-{-b,  added  to  c-\-d,  is  a-\-b-{-c-\-d.  In  the  same  manner,  if 
the  sum  of  any  quantities  whatever,  be  added  to  the  sum  of 


22  ALGEBRA. 

any  others,  the  expression  for  the  whole,  will  contain  all 
these  quantities  connected  by  the  sign  -)-• 

67.  Again,  if  the  difference  of  a  and  b  be  added  to  c ;  the 
sum  will  be  a- 6  added  to  c,  that  is  a-h-\-c.  And  ii a-h 
be  added  to  c-d,  the  sum  will  be  a-h-\-c-d.  In  one  of 
the  compound  quantities  added  here,  a  is  to  be  diminished 
by  6,  and  in  the  other,  c  is  to  be  diminished  by  d ;  the  sum 
of  a  and  c  must  therefore  be  diminished,  both  by  5,  and  by 
d,  that  is,  the  expression  for  the  sum  total,  must  contain  -  6 
and  -d.  On  the  same  principle,  all  the  quantities  which,  in 
the  parts  to  be  added,  have  the  negative  sign,  must  retain  this 
sign  in  the  amount.  Thus  «-[-26-c,  added  to  c?-/i  m,  is 
a-\-U-c-]-d-k-m. 

68.  The  sign  must  be  retained  also,  when  a  positive  quan- 
tity is  to  be  added,  to  a  single  negative  quantity.  If  a  be 
added  to  -  6,  the  sum  will  be  -  h-]-a.  Here  it  may  be  object- 
ed, that  the  negative  sign  prefixed  to  6,  shows  that  it  is  to  be 
subtracted.  What  propriety  then  can  there  be  in  adding  it  1 
In  reply  to  this,  it  may  be  observed,  that  the  sign  prefixed 
to  b  while  standing  alone,  signifies  that  b  is  to  be  subtracted, 
not  from  a,  but  from  some  other  quantit}^,  which  is  not  here 
expressed.  Thus  -b  may  represent  the  loss,  which  is  to  be 
subtracted  from  the  stock  in  trade.  (Art.  55.)  The  object 
of  the  calculation,  however,  may  not  require  that  the  value 
of  this  stock  should  be  specified.  But  the  loss  is  to  be  con- 
nected with  a  profit  on  some  other  article.  Suppose  the 
profit  is  2000  dollars,  and  the  loss  400.  The  inquiry  then,  is 
what  is  the  value  of  2000  dollars  profit,  when  connected  with 
400  dollars  loss  ? 

The  answer  is  evidently  2000-400,  which  shows  that 
2000  dollars  are  to  be  added  to  the  stock,  and  400  subtracted 
from  it ;  or  which  will  amount  to  the  same,  that  the  difference 
between  2000  and  400  is  to  be  added  to  the  stock. 

69.  Quantities  are  added,  then,  by  writing  them  one 

AFTER  another,  WITHOUT  ALTERING  THEIR  SIGNS  ;  observ- 
ing always,  that  a  quantity,  to  Avhich  no  sign  is  prefixed,  is 
to  be  considered  positive.      (Art.  29.) 

The  sum  of  a-\-m,  and  6-8,  and  2/i-3m-(-<^,  and  /i-n 
and  r-[-3m-i/,  is 

a~\-m-\-b  -  8-f  2/i  -  Sm+cZ-j-Zi  -  n-j-r-f-Sm  -  y. 

70.  It  is  immaterial  in  what  order  the  terms  are  arranged. 
The  sum  of  a  and  b  and  c  is  either  a-j-6-|-c,  or  a-\-c-\-b,  or 
c-\-b-\-a.  For  it  evidently  makes  no  diflference,  which  of  the 
quantities  is  added  first.  The  sum  of  6  and  3  and  9,  is  the 
same  as  3  and  9  and  6,  or  9  and  6  and  3. 


ADDITION.  23 

And  a-\-m  -  n,  is  the  same  as  a  -  n~\-m.  For  it  is  plainly 
of  no  consequence,  whether  Ave  first  add  m  to  a,  and  after- 
wards subtract  n;  or  first  subtract  n  and  then  add  m. 

71.  Though  connecting  quantities  by  their  signs  is  all 
which  is  essential  to  addition  ;  yet  it  is  desirable  to  make  the 
expression  as  simple  as  ma}^  be,  by  reducing  several  terms  to 
one.     The  amount  of  Sa,  and  6b,  and  4«,  and  56,  is 

3a+66+4ft+56. 
But  this  may  be  abridged.  The  first  and  third  terms  may 
be  brought  into  one;  and  so  may  the  second  and  fourth. 
For  3  times  a,  and  4  times  a,  mai^e  7  times  a.  And  6  times 
h,  and  5  times  6,  make  1 1  times  b.  The  sum  when  reduced 
is  therefore  7a-\-l\b. 

For  making  the  reductions  connected  with  addition,  two 
rules  are  given,  adapted  to  the  two  cases,  in  one  of  which, 
the  quantities  and  signs  are  alike,  and  in  the  other,  the  quan- 
tities are  alike,  but  the  signs  are  vmlike.  Like  quantities 
are  the  same  powers  of  the  same  letters.  (Art.  45.)  But 
as  the  addition  of  powers  an^  radical  quantities  will  be  con- 
sidered in  a  future  section,  the  examples  given  in  this  place, 
will  be  all  of  the  first  power. 

72.  Case  I.     To  reduce  several  teRxMS  to  one,  when 

THE  QUANTITIES  ARE  ALIKE,  AND  THE  SIGNS  ALIRE,  ADD  THE 
co-efficients,  annex  THE  COMMON  LETTER  OR  LETTERS, 
AND  PREFIX  THE  COMMON  SIGN. 

Thus  to  reduce  36+76,  that  is  -f  36+76  to  one  term,  add 
the  co-efficients  3  and  7  ;  to  the  sum  10,  annex  the  common 
letter  6,  and  prefix  the  sign  +.  The  expression  will  then 
be  -|-106.  That  3  times  any  quantity,  and  7  times  the  same 
quantity,  make  10  times  that  quantity,  needs  no  proof. 


Examples. 


be 

Sxy 

76+  xy 

ry-]-Sabh 

2bc 

Ixy 

86+3a:i/ 

Snj-\-  abh 

96c 

ccy 

2bJf-2xy 

6ry-\-4abh 

36c 

2xy 

66+5.11/ 

2ry-{-  abh 

cdxy-\-Smg 

2cdocy-\-  mg 
5cdxy-\-7mg 
'7cdxy-\-8mg 


156c        '  236+1  la^  {5cdxy-{.l9mg 

The  mode  of  proceeding  will  be  the  same,  if  tlie  signs  are 
egative. 
Thus  -  36c  -  6c  -  56c,  becomes,  when  reduced,  -  96c. 


24  ALGEBRA. 

And  -  ax  -  Sax  -  2ax=-  6ax.     Or  thus, 

-  36c  -   ax  '  -  2ab  -   my  -  Sack  -  Sbdy 

~    be  -Sax  -    ab-Smy  -   ach-    bdy 

-obc  -2ax  -7ab-Smy  -5ach-7bdy 


-96c  -Wab-n 


my 


73.  It  may  perhaps  be  asked  here,  as  in  art.  68,  what  pro- 
priety there  is,  in  adding  quantities,  to  which  the  negative 
sign  is  prefixed  ;  a  sign  which  denotes  subtraction  ?  The  an- 
swer to  this  is,  that  when  the  negative  sign  is  apphed  to  sev- 
eral quantities,  it  is  intended  to  indicate  that  th  ^se  quantities 
are  to  be  subtracted,  not  from  each  other,  but  from  some  other 
quantity  marked  wilh  the  contrary  sign.  Suppose  that,  in 
estimating  a  man's  propert}^,  the  sum  of  money  in  his  pos- 
session is  marked  -j-,  and  the  debts  which  he  owes  are  mark- 
ed-. If  these  delMs  are  200,  300,  500  and  700  dollars,  and 
if  a  is  put  for  100 ;  they  will  together  he  -2a- Sa- 5a-  la. 
And  the  several  terms  reduced  to  one,  will  evidently  be 
-17a,  that  is,  1700  dollars. 

74.  Case  II.    To  reduce  several  terms  to  one,  when 

THE  quantities  ARE  ALIKE,  BUT  THE  SIGNS  UNLIKE,  TAKE 
THE  LESS  CO-EFFICIEKT  FROM  THE  GREATER;  TO  THE  DIF- 
FERENCE, ANNEX  THE  COMMON  LETTER  OR  LETTERS,  AND 
PREFIX  THE  SIGN  OF  THE  GREATER  CO-EFFICIENT. 

Thus,  instead  of  8a -6a,  we  may  write  2a. 

And  instead  of  76-26,  we. may  put  56. 

For  the  simple  expression,  in  each  of  these  instances,  is 
equivalent  to  the  compound  one  for  which  it  is  substituted. 
To  -1-66       4-46       56c         2/im      -dy-{-Qm     Sh-    dx 

Add  -46        -66   -76c      -96?n      Ady  -   m     bh-^Adx 


Sum+26  -26c  Sdy-\-5m 


75.  Here  again,  it  may  excite  surprise,  that  what  appears 
to  be  subtraction,  should  be  introduced  under  addition.  But 
according  to  what  has  been  observed,  (Art.  66.)  this  subtrac- 
tion is  strictly  speaking,  no  part  of  the  addition.  It  belongs 
to  a  consequent  reduction.  Suppose  66  is  to  be  added  to 
a -46.     The  sum  is  a -46+66.     (Art.  69.) 

But  this  expression  may  be  rendered  more  simple.  As  it 
now'  stands,  46  is  to  be  subtracted  from  a,  and  66  added. 
But  the  amount  will  be  the  same,  if,  without  subtracting  any 
thing,  we  add  26,  making  the  whole  a-1-26.     And  in  all  sim- 


ADDITION.  25 

ilar  instances,  the  balance  of  two  or  more  quantities,  may  be 
substituted  for  the  quantities  themselves. 

77.  If  two  equal  quantities  have  contrary  signs,  they  de- 
stroy each  otlier,  and  may  be  cancelled.  Thus -{-66-66 
=  0:  And  3x6-183=0:  And  76c -7^  =  0. 

Let  there  be  any  two  quantities  wliatever,  of  which  a  is 
the  greater,  and  6  the  less. 

Their  sum  will  be         a-|-6 
And  their  difference      a-b 


The  sum  and  difference  added,  will  be  2a-[-0,  or  simply 
2a.  That  is,  if  the  sum  and  difference  of  any  two  quantities 
be  added  together,  the  ichole  will  be  twice  the  greater  quan- 
tity. This  is  one  instance,  among  multitudes,  of  the  rapidity 
with  \\'hich  general  truths  are  discovered  and  demonstrated 
in  algebra.     (Art.  23.) 

78.  If  several  positive,  and  several  negative  quantities  are 
to  be  reduced  to  one  term  ;  first  reduce  those  which  are  posi- 
tive, next  those  which  are  negative,  and  then  take  the  differ- 
ence of  the  co-efficients,  of  the  two  terms  thus  found. 
Ex.  1.  Reduce  136-1-664  b- 4b -5b -7b,  to  one  term. 
By  art.  72,  136-f-66+  6^     206  > 

166  5 


And  -46-56-76: 


By  art.  74,  206-166  =  46,  which  is  the  value 

of  alJ  the  given  quantities,  taken  together. 

Ex.  2.  Reduce  Sxy  -  xy-\-2xij  -  7xy-\-4xy  -  9xy-\-7xy  -  6xy. 

The  positive  terms  are  Sxy     The  negative  terms  are  -  xy 

2xy  ^7xy 

4xy  -9xy 

7xy  ~6ari/ 

And  their  sum  is        16xy  -2Sxy 

Then  l6ory-2Sxy=-7xy 

Ex.  3.  Sad-ead-^ad-{-7ad-2ad^9ad-8ad-4ad=0. 

4.  2abm  -  abm-\-7abm  -  3«6m-j-7a6m= 

5.  axy-7axy-\-8ao(ry-axy-Sa3cy-\-9a3n/= 

79.  If  the  letters,  in  the  several  terms  to  be  added,  are 
different,  they  can  only  be  placed  after  ea  :h  other,  with  their 
proper  signs.     They  cannot  be  united  in  one  simple  term. 


26  ALGEBRA. 

If  4b,  and  -  6j/,  and  Sx,  and  17 h,  and  -  5d,  and  6,  be  added ; 
their  sum  will  be 

4b-6y+Sx+\7h-5d-\-6.   (Art.  69.) 

Different  letters  can  no  more  be  united  in  the  same  term, 
than  dollars  and  guineas  can  be  added,  so  as  to  make  a 
single  sum.  Six  guineas  and  4  dollars  are  neither  ten  guineas 
nor  ten  dollars.  Seven  hundred  and  five  dozen,  are  neither 
12  hundred  nor  12  dozen.  But,  in  such  cases,  the  algebraic 
signs  serve  to  show  how  the  different  quantities  stand  related 
to  each  other ;  and  to  indicate  future  operations,  which  are 
to  be  performed,  whenever  the  letters  are  converted  into 
numbers.  In  the  expression  «-]-6,  the  two  terms  cannot  be 
united  in  one.  But  if  a  stands  for  15,  and  if,  in  the  course 
of  a  calculation,  this  number  is  restored ;  then  a-\-6  will  be- 
come 15-[-6,  which  is  equivalent  to  the  single  term  21.  In 
the  same  manner,  a -6,  becomes  15-6,  which  is  equal  to  9. 
The  signs  keep  in  view  the  relations  of  the  quantities  till  an 
opportunity  occurs  of  reducing  several  terms  to  one. 

80.  When  the  quantities  to  be  added  contain  several  terms 
which  are  alike,  and  several  which  are  unlike,  it  will  be  con- 
venient to  arrange  them  in  such  a  manner,  that  the  similar 
terms  may  stand  one  under  another. 

To       36c  -  6d-\-2b  -Sy  ^      These  may  be  arranged  thus : 

Add  -Sbc-\-x-Sd-{-bg    >      Sbc-6d-\-2b  -  Sy 

And     2d-{-y-\-^x+b       )   -Shc-Sd  +  x-\-bg 

2d  -f  y^Sx  4-6 


The  sum  will  be  -7d-^2b-  2yJ^4x-\-bg-{- b. 

Examples. 

1.  Add  and  reduce  ab-\-8  to  cd-S  and  5ab  -  4in-\-2, 
The  sum  is  6ab-\-7-\-cd-4m. 

2.  Add  x^2ty-dx,  to  7 -x-8-\-hm. 
Ans.  ^y-dx-\-\-h;m. 

3.  Add  abm-^x-\-bm,  to  y-x-\-7  and  5x-Qy-\-^. 

4.  Add  3am-f  6-7a:t/-8,  to  10:i^-9+5am. 

5.  Add  Qahy-\-7d-\-\-mxy,  to  Zahy-7d-\-\7 -mxy 

6.  Add  7 ad  -  h-\-8xy  -  ad,  to  5ad--\~h  -  7xy, 

7.  Add  3a6  -  2ay-\-x,  to  ab  -  ay-\-bx  -  h. 
8    Add  2by  -  Sax-\-2a,  to  Sbx  -  %+a. 


SUBTRACTION.  2T, 


SECTION  III, 


SUBTRACTION. 

Arx,  81.  ADDITION  is  bringing  quantities  together,  to 
find  their  amount.     On  the  contrary,  SUBTRACTION  is 

FINDING  THE  DIFFERENCE  OF  TWO  QUANTITIES,  OR  SETS 
OF  QUANTITIES. 

Particular  rules  might  be  given,  for  the  several  cases  in 
subtraction.  But  it  is  more  convenient  to  have  one  general 
rule,  founded  on  the  principle,  that  taking  aioay  a  positive 
quantity,  from  an  algebraic  expression,  is  the  same  in  effect, 
as  annexing  an  equal  negative  quantity ;  and  taking  away 
a  negative  quantity  is  the  same,  as  annexing  an  equal  posi- 
tive one. 

Suppose  -^h  is  to  be  subtracted  from  a-\-h 

Taking  away  -|-^)  ^o"^  «H-^j  leaves  a 

And  annexing  -h,  to  a-{-h,  gives  a-\-h-h 

But  by  axiom  5th,  a-\-h  -  6  is  equal  to  a 

That  is,  taking  away  a  positive  term,  from  an  algebraic 
expression,  is  the  same  in  effect,  as  annexing  an  equal  nega- 
tive term. 

Again,  suppose  - 6  is  to  be  subtracted  from     a-b  • 

Taking  away  -  6,  from  a-b,  leaves  a 

And  annexing  -\-b,  to  a -6,  gives  a-b-\-b 

But  a-b-{-b  is  eiqual  to  a 

That  is,  taking  away  a  negative  term,  is  equivalent  to  an- 
nexing a  positive  one.  If  an  estate  is  encumbered  with  a 
debt ;  to  cancel  this  debt  is  to  add  so  much  to  the  value  of 
the  estate.  Subtracting  an  item  from  one  side  of  a  book  ac- 
count, will  produce  the  same  alteration  in  the  balance,  as 
adding  an  equal  sum  to  the  opposite  side. 
To  place  this  in  another  point  of  view. 
If  m  is  added  to  6,  the  sum  is  by  the  notation  6-fm  > 

But  if  m  is  subtracted  from  b,  the  remainder  is    6  -  w  3 
So  if  m  and  h  are  each  added  to  b,  the  sum  is      b-\-m-\-h  ^ 
But  if  m  and  h  are  each  subtracted  from  6,  the  > 

remainder  is  b-m-h  ) 


^>8  ALGEBRA. 

The  only  difference  then  between  addhig  a  positive  quan- 
tity and  subtracting  it,  is,  that  the  dgn  is  changed  from  -{- 
to-. 

Again,  if  m-n  is  subtracted  from  b,  the  remainder  is, 
b  —  m-\-n. 
For  tJie  less  the  quantity  subtracted,  the  greater  will  be  the 
remainder.  But  in  the  expression  m-n,  m  is  diminished  by 
n;  therefore,  b  —  m  must  be  increased  b}'^  n;  so  as  to  become 
b-m-\-n:  that  is,  m-n  is  subtracted  from  fe,  by  changing 
-}-wi  into  -m,  and  -n  into  -j-**j  ^^"^^  ^^^^i^  writing  them  after 
5,  as  in  addition.  The  explanation  will  be  the  same,  if  there 
are  several  quantities  which  have  the  negative  sign.     Hence, 

82.  To  PERFORM  SUBTRACTION  IN  ALGEBRA,  CHANGE  THE 
SIGNS  OF  ALL  THE  QUANTITIES  TO  BE  SUBTRACTED,  OR  SUP- 
POSE THEM  TO  BE  CHANGED,  FROM  -j-  TO  -,  OR  FROM  -  TO  -[-, 
AND  THEN  PROCEED  AS  IN  ADDITION. 

The  signs  are  to  be  changed,  in  the  subtrahend  only. 
Those  in  the  minuend  are  not  to  be  altered.  Although  tlie 
rule  here  given  is  adapted  to  every  case  of  subtraction ;  yet 
there  may  be  an  advantage  in  giving  some  of  the  examples 
in  distinct  classes. 

83.  In  the  first  place,  the  signs  may  be  alike,  and  the  min- 
uend greater  than  the  subtrahend. 

From  -f28       166       Uda      -28       -166       -Uda 

Subtract         ^16        126         Qda       -16       -126         ~Qda 


Pifference  +12  46  Sda  -12  -46  -Ma 
Here,  in  the  first  example,  the -|- before  16  is  supposed 
to  be  changed  into  -,  and  then,  the  signs  being  unlike,  the 
two  terms  are  brought  into  one,  by  the  second  case  of  re- 
duction in  addition.  (Art.  74.)  The  two  next  examples 
are  subtracted  in  the  same  ^va)^  In  the  three  last,  the  -  in 
the  subtrahend,  is  supposed  to  be  changed  into  +.  It  may 
be  well  for  the  learner,  at  fiist,  to  write  out  the  examples ; 
and  actually  to  change  the  signs,  instead  of  merely  con- 
ceiving them  to  be  changed.  When  he  has  become  familiar 
with  the  operation,  he  can  save  himself  the  trouble  of  tran- 
scribing. 

This  case  is  the  same  as  subtraction  in  arithmetic.     The 
two  next  cases  do  not  occur  in  common  arithmetic. 

84.  In  the  second  place,  the  signs  may  be  alike,  and  the 
minuend  less  than  the  subtrahend. 


From 

Sub. 

-fl66 

4-286 

-126 

SUBTRACTION. 

126         6da         -16 
166       \4da         -28 

-126 
-166 

29 

-   6da 
-14rfa 

Dif. 

-46      -Sda       4-12 

46 

8da 

The  same  quantities  are  given  here,  as  in  the  preceding 
article,  for  the  purpose  of  comparing  them  together.  But  the 
minuend  and  subtrahend  are  made  to  change  places.  The 
mode  of  subtracting  is^  tlie  same.  In  this  class,  a  greater 
quantity  is  taken  from  a  less  :  in  the  preceding,  a  less  from  a 
greater.  By  comparing  them,  it  will  he  seen,  that  there  is  no 
difference  in  the  answers,  except  that  the  signs  are  opposite. 
Thus  166-  126  is  the  same  as  126-  166,  except  that  one  is 
4-46,  and  the  other  -  46 ;  That  is,  a  greater  quantity  sub- 
tracted from  a  less,  gives  the  same  result,  as  a  less  subtracted 
from  a  greater,  except  that  the  one  is  positive,  and  the  other 
negative.     See  Art.  58  and  59. 

85.  In  the  thiid  place,  the  signs  may  be  unlike. 

From      4-28       4-166       -\-^4da       -28       -166       -\4da 
Sub.         -16        -126        -    6da      4-16      4-126      4-  6da 


Dif.         4-44  286  20da     -44      -286       -20da 

From  these  examples,  it  will  be  seen  that  the  difference 
between  a  positive  and  a  negative  quantity,  may  be  greater 
than  either  of  the  two  quantities.  In  a  thermometer,  the  dif- 
ference between  28  degrees  above  cypher,  and  16  below,  is 
44  degi'ees.  The  difference  between  gaining  1000  dollars  in 
trade,  and  losing  500,  is  equivalent  to  1500  dollars. 

86.  Subtraction  may  be  proved,  as  in  arithm.etic,  by  adding 
the  remainder  to  the  subtrahend.  The  sum  ought  to  be  equal 
to  the  minuend,  upon  the  obvious  principle,  that  the  difference 
of  two  quantities  added  to  one  of  them,  is  equal  to  the  other. 
This  serves  not  only  to  correct  any  particular  error,  but  to 
verify  the  general  rule. 

From       2a:i/-l  /i4-36a;  hy-    ah  nd-7by 

Sub.        -xy+7      .Sh-9bx  5hy^6ah  5nd-   by 

Dif.  Sxy-S        -^^^O^'^"       _4%4-5a^ 

From     Sabm-   xy         -174-4rta:  ax-\-  76        Sah-\-'axy 

Sub.  -7abm-\-6xy         -20-   ax      -4ax-\-\5b.    -7ah-{-axy 

Rem.  \0abm--7xy  5ax-   86 


30  ALGEBRA. 

87.  When  there  are  several  terms  alike,  they  may  be  re- 
duced as  in  addition. 

1.  From  ab  subtract  Sam-{-am~{-7am-\-2am-\-6ain. 

Ans.  ab  -  Sam  -  am  -  lam  -2am-  6am =ab-l  9am.   (Art.  72. ) 

2.  From  ?/,  subtract -a-a-a-a. 
Ans.  y-\-a-\-a-\-a-\-a=y-\-4a. 

3.  From  ax  -  bc-\-Sax-]-7bc,  subtract  4bc-2ax-{-bc-\-4ax. 
Ans.  ax  -  bC'\-Sax-\-7bc  -  4bc-\-2ax  -be-  4ax  —  2ax-\-bc. 
(Art.  78.) 

4.  From  ad-{-Sdc-bx,  subtract  Sad-\-7bx  -  dc-\-ad. 

88.  When  the  letters  in  the  minuend  are  different  from 
tliose  in  the  subtrahend,  the  latter  are  subtracted,  by  first 
changing  the  signs,  and  then  placing  the  sev^eral  terms  one 
after  another,  as  in  addition.   (Art.  79.) 

From  Sab-\-S-my-\-dhy  subtract  x  -  dr-\-4hy  -bmx. 
Ans.  Sab-{S  -  my-\-dh  -  x-{-dr  -  4hy-\-bmx. 

88.  b.  The  sign  - ,  placed  before  the  marks  of  parenthesis, 
which  include  a  number  of  quantities,  requires,  that  when 
these  marks  are  removed,  the  signs  of  all  the  quantities  thus 
included,  should  be  changed. 

Thus  a-  {b-c-{-d)  signifies  that  the  quantities  b,  ~c,  and 
-f-d,  are  to  be  subtracted  from  a.  The  expression  Avill  then 
become  a-b-\-c-d. 

2.  \Sad-\-xy-\-d-  (7ad-ocy-\-d-\-hm-ry)  =  6ad-\-2xy  -hm 

3.  7abc-8-{-7x-  {Sabc-8-dx+r)=4abc-\-'7x-\-dx-r. 

4.  Sad-\-h  -2y-  {7y-\-Sh  -  mx-\-4ad  -hy-  ad)  = 

5.  6am'-dy-\-8-{l6-{-Sdy-8-{-am-e-{-r)  = 

6.  7ay-2x-\-5-{4-[-h-ay+x-I^Sb)  = 

88.  c.  On  the  other  hand,  when  a  number  of  quantities  are 
introduced  within  the  marks  of  parenthesis,  with  -  immedi- 
ately preceding;  the  signs  must  be  changed. 

Thus  -m+b-dx+Sh=.-{m-b+dx-Sh.) 


MULTIPLICATION.  31 


SECTION  IV. 


MULTIPLICATION.* 

Art.  89.  In  addition,  one  quantity  is  connected  with  an- 
other. It  is  frequently  the  case,  that  the  quantities  brought 
together  are  equal ;  that  is,  a  quantity  is  added  to  itself. 

As  a-\-a=i2a  a-\-a-{-a-\-a=i4a 

a-\-a-\-a=Sa  a^a-\-a-\-a-\-a=5a,  &c. 

This  repeated  addition  of  a  quantity  to  itself,  is  what  was 
originally  called  multiplication.  But  the  term,  as  it  is  now 
used,  has  a  more  extensive  signification.  We  have  frequent 
occasion  to  repeat,  not  only  the  ichole  of  a  quantity,  but  a 
certain  portion  of  it.  If  the  stock  of  an  incorporated  com- 
pany is  divided  into  shares,  one  man  may  own  ten  of  them, 
another  five,  and  another  a  part  only  of  a  share,  say  two- 
fifths.  When  a  dividend  is  made,  of  a  certain  sum  on  a 
share,  the  first  is  entitled  to  ten  times  this  sum,  the  second  to 
five  times,  and  the  third  to  only  tino-fifths  of  it.  As  the  ap- 
portioning of  the  dividend,  in  each  of  these  instances,  is 
upon  the  same  principle,  it  is  called  multiplication  in  the 
last,  as  well  as  in  the  two  first. 

Again,  suppose  a  man  is  obligated  to  pay  an  annuity  of  100 
dollars  a  year.  As  this  is  to  be  subtracted  trom  his  estate,  it 
may  be  represented  by  -a.  As  it  is  to  be  subtracted  year 
after  year,  it  will  become,  in  four  years,  -a-a-a- a=  - 4a. 
This  repeated  subtraction  is  also  called  multiplication.  Ac- 
cording to  the  view  of  the  subject; 

90.  Multiplying  by  a  whole  number  is  taking  the 
multiplicand  as  many  times,  as  there  are  units  in  the 
multiplier. 

Multiplying  by  1,  is  taking  the  multiplicand  once,  as  a. 
Multiplying  by  2,  is  taking  the  multiplicand  twice,  as  a-\-a. 


*  Newton's  Universal  Arithmetic,  p.  4.  Maseres  on  the  Negative  Sign, 
Sec.  II.  Camus'  Arithmetic,  Book  II.  Chap.  3.  Euler's  Algebra,  Sec.  I 
II.  Chap.  3.  Simpson's  Algebra,  Sec.  IV.  Maclaurin,  Saunderson,  Lacroix, 
Ludlam. 


32  ALGEBRA. 

Multiplying  by  3,  is  taking  the  multiplicand  three  times^  as 
a-\-a-\-a,  &c. 

Multiplying  by  a  FRACTION  is  taking  a  certain 
PORTION  OF  THE  multiplicand  as  many  times,  as  there 
are  like  portions  of  a  unit  in  the  multiplier.* 
Multiplying  by  ^j  is  taking  \  of  the  multiplicand,  once,  as  \a. 
Multiplying  by  |,  is  taking  \  of  the  multiplicand,  twice,  as 

\a-\-xa. 
Multiplying  by  |,  is  taking  ^  of  the  multiplicand,  three  times. 

Hence,  if  the  multiplier  is  a  unit,  the  product  is  equal  to 
the  multiplicand  :  If  the  multiplier  is  greater  than  a  unit,  the 
product  is  greater  than  the  multiplicand :  And  if  the  multipli- 
er is  less  than  a  unit,  the  product  is  less  than  the  multiplicand. 

Multiplication  by  a  NEGATIVE  quantity,  has  the 
same  relation  to  multiplication  by  a  positive  quantity, 
which  SUBTRACTION  has  to  addition.  In  the  one,  the 
sum  of  the  repetitions  of  the  multiplicand  is  to  be  added,  to 
the  other  quantities  with  which  this  muiCiplier  is  connected. 
In  the  other,  the  sum  of  these  repetitions  is  to  be  subtracted 
from  the  other  quantities.  This  subtraction  is  performed  at 
the  time  of  multiplying,  by  changing  the  sign  of  the  pro- 
duct.    See  Art.  107  and  108. 

91.  Every  multiplier  is  to  be  considered  a  number.  We 
sometimes  speak  of  multiplying  bj^a  given  weight  or  measure, 
a  sum  of  money,  &c.  But  this  is  abbreviated  language.  If 
construed  literally,  it  is  absurd.  Multiplying  is  taking  either 
the  whole  or  a  part  of  a  quantity,  a  certain  number  of  times. 
To  say  that  one  quantity  is  repeated  as  many  times,  as  an- 
other is  heavy,  is  nonsense.  But  if  a  part  of  the  w  eight  of  a 
body  be  fixed  upon  as  a  unit,  a  quantity  may  be  multiplied 
by  a  number  equal  to  the  number  of  these  parts  contained 
in  the  body.  If  a  diamond  is  sold  by  weight,  a  particulai 
price  may  be  agreed  upon  for  each  grain.  A  grain  is  here 
the  unit ;  and  it  is  evident  that  the  value  of  the  diamond,  is 
equal  to  the  given  price  repeated  as  many  times,  as  there  are 
grains  in  the  whole  weight.  We  say  concisely,  that  the  price 
is  multiplied  by  the  weight ;  meaning  that  it  is  multiplied  by 
a  number  equal  to  the  number  of  grains  in  the  weight.  In  a 
similar  manner,  any  quantity  whatever  may  be  supposed  to 
be  made  up  of  parts,  each  being  considered  a  unit,  and  any 
nmiiber  of  these  may  become  a  multiplier. 


*  See  Note  C. 


MULTIPLICATION.  33 

92.  As  multiplying  is  taking  the  whole  or  a  part  of  a 
quantity  a  certain  number  of  times,  it  is  evident  that  the 
product,  must  be  of  the  same  nature  as  the  multiplicand. 

If  the  multiplicand  is  an  abstract  number;  the  product  will 
be  a  number. 

If  the  multiplicand  is  weight,  the  product  will  be  weight. 
If  the  multiplicand  is  a  line,  the  product  will  be  a  line.  Re- 
peating a  quantity  does  not  alter  its  nature.  It  is  frequently 
said,  that  the  product  of  two  lines  is  a  surface,  and  that  the 
product  of  three  lines  is  a  solid.  But  these  are  abbreviated 
expressions,  \Yhich  if  interpreted  literally  are  not  correct. 
See  Section  xxi. 

93.  The  multiphcation  of  fractions  will  be  the  subject  of 
a  future  section.  We  have  first  to  attend  to  multiplication 
by  positive  whole  numbers.  This,  according  to  the  defini- 
tion (Art.  90.)  is  taking  the  midtiplicand  as  many  times,  as 
there  are  units  in  the  multiplier.  Suppose  a  is  to  be  multi- 
plied by  b,  and  that  b  stands  for  3.  There  are  then,  three 
units  m  the  multiplier  b.  The  multiphcand  must  therefore 
be  taken  three  times;  thus,  a-}-a-{-a=Sa,  or  ba. 

So  that,  multiplying  two  letters  together  is  nothing  more, 
than  writing  them  one  after  the  other,  either  with,  or  without 
the  sign  of  multiplication  between  them.  Thus  b  multiphed 
into  CIS  bxc,  or  be.     And  x  into  y,  is  ccXy,  or  x.y,  or  on/. 

94.  If  more  than  tw^o  letters  are  to  be  multiplied,  they 
must  be  connected  in  the  same  manner.  Thus  a  into  b  and 
c,  is  cba.  For  by  the  last  article,  a  into  b,  is  ba.  This  pro- 
duct is  now  to  be  multiplied  into  c.  If  c  stands  for  5,  then 
ba  is  to  be  taken  five  times  thus, 

ba-{-ba-\-ba-{-ba-\-ba=5ba,  or  cba. 
The  same  explanation  may  be  applied  to  any  number  of 
letters.     Thus,  am  into  xy,  is  amxy.     And  bh  inio  mrx,  is 
bhmrx. 

95.  It  is  immaterial  in  what  order  the  letters  are  arranged. 
The  product  ba  is  the  same  as  ab.  Three  times  five  is  equal 
to  five  times  three.  Let  the  nmiiber  5  be  represented  by  as 
many  points,  in  a  horizontal  line  ;  and  the  number  3,  by  as 
many  points  in  a  perpendicular  line. 


Here  it  is  evident  that  the  whole  number  of  points  is  equal, 
either  to  the  number  in  the  horizontal  row  three  times  repeat- 

5 


34  ALGEBRA. 

ed,  or  to  the  number  in  the  perpendicular  row  Jive  times  re- 
peated ;  that  IS,  to  5x3,  or  3x5.  This  explanation  may  be 
extended  to  a  series  of  factors  consisting  of  any  numbers 
whatever.  For  the  product  of  two  of  the  factors  may  be 
considered  as  one  number.  This  may  be  placed  befoie  or 
after  a  third  factor  :  the  product  of  three,  before  or  after  a 
fourth,  &c. 

Thus  24=4x6  or  6x4  =  4x3x2  or  4x2x3  or  2x3x4. 

The  product  of  a,  b,  c,  and  d,  is  abed,  or  acdb,  or  dcba,  or  bade. 
It  will  generally  be  convenient,  however,  to  place  the  letters 
in  alphabetical  order. 

96.  When  the  letters  have  numerical  CO-EFFI- 
CIENTS, THESE  MUST  BE  MULTIPLIED  TOGETHER,  AND 
PREFIXED    TO    THE    PRODUCT    OF    THE    LETTERS. 

Thus,  3rt  into  2b,  is  6ab.  For  if  a  into  b  is  ab,  then  3  times 
a  into  b,  is  evidently  Sab :  and  if,  instead  of  multiplying  by 
h,  we  multiply  by  twice  b,  the  product  must  be  ticice  as  great ; 
that  is  2x3a6  or  6ab. 

Mult.  9ab         12%         Sdh         2ad  7bdh         Say 

Into  Sxy  2rx  my      IShmg  x  ^mx 


Prod.       21abxy  Sdhmy  7bdhx 

97.  If  either  of  the  factors  consists  of  figures  only,  these 
must  be  multiplied  into  the  co-efficients  and  letters  of  the 
other  factors. 

Thus  Sab  into  4,  is  \2ab.  And  36  into  2x,  is  72a:.  And 
24  into  hy,  is  24%. 

98.  If  the  multiplicand  is  a  compound  quantity,  each  of  its 
terms  must  be  multiplied  into  the  multiplier.  Thus  b-\-c-\-d 
into  a  is  ab-\-ac-\-ad.  For  the  whole  of  the  multiplicand  is 
to  be  taken  as  many  times,  as  there  are  units  in  the  multi- 
plier. If  then  a,  stands  for  3,  the  repetitions  of  the  multipli- 
cand are, 

bJ[-c^d 
b-}-c-\-d 


And  their  sum  is       Sb-\-Sc-{-Sd,  that  is,  ab-\-a-c-}-ad. 


MULTIPLICATION.  35 

Malt.  d-{-^xy         2/i-fm  3/iZ+l  2/im+3+£/r 

Into  36  6dy  my  4b 


Prod        SbdJ^Gbxy  Shhmj-\-my 


99.  The  preceding  instances  must  not  be  confounded 
with  tliose  in  which  several  factors  are  connected  by  the 
sign  X J  or  by  a  point.  Li  the  latter  case,  the  multipHer  is 
to  be  written  before  the  other  factors  without  being  rcpeatCil. 
The  product  of  bxd  into  a,  is  ab  x  d,  and  not  abx^id-  For 
bxd  is  bd,  and  this  into  a,  is  abd.  (Art.  94.)  The  expression 
bxd  is  not  to  be  considered,  like  b+d,  a  compound  quantify 
consisting  of  two  terms.  Different  terms  are  always  separ-i- 
ted  by+or-.  (Art.  36.)  The. product  of  bx^X^^^Xy  h^- 
to  a,  is  ax^Xhx>^iXy  oy  abhmy.  But  b-i^h-\-m-\-y  into  a, 
is  ab-{-ah-{-am-\-ay. 

100.  If  both  the  factors  are  compound  quantities,  each 
term  in  the  multiplier  must  be  multiplied  into  each  in  the  multi- 
plicand. 

Thus  a-\-b  into  c-{-d  is  ac-{-ad-\-bcJ^bd. 

For  the  units  in  the  multiplier  a-\-b  are  equal  to  the  units 
in  a  added  to  the  units  in  b.     Therefore  the  product  produ- 
ced by  a,  must  be  added  to  the  product  produced  by  b. 
The  product  of  c-{-d  into  a  is  ac-{-ad  >        »       go 
The  product  of  c-\-d  into  b  is  bc-\-bd  )       ^     '       ' 
The  product  of  c-\-d  into  a-j-5  is  therefore  ac-]-ad-\-bc-\-bd. 

MuU.     SxJ^d  4ay^2b         a+1 

Into        2a-\-hm  3c  -\-rx        3a:-f  4 


Prod.  6ax-{-2ad-\-^hmx-irdhm  3aa:_|_3a;+4a-f  4 

Mult.  2/1+7  into  6^+1.     Prod.  \2dh-{-42d-[-2h-\-l. 
Mult.  dy-\-rx-\-h  into  6»i+4+7i/.     Prod. 
Mult.  7+66+af^  into  3r+4+2/i.     Prod. 

101.  When  several  terms  in  the  product  are  alike,  it  will 
be  expedient  to  set  one  under  the  other,  and  then  to  unite 
them,  by  the  rules  for  the  reduction  in  addition. 


36  ALGEBRA. 

Mult.       b-\-a  6+C-J-2  a+  y+l 

Into  6+a  i+c+3  36+2a:+7 

J^ab-\-aa        '  he         4-cc-f-2c  * 

_j_36        +3c+6 


Prod.         hh-\-^ah-\-aa  66+2k+56+cc+5c4-6 


Mult.  3a+6^4-4  into  2a+3rf4-].  Prod. 
Mult.  h-\-cd^2  into  3fe-|-4crf-|-7.  Prod. 
Mult.       36+2a;+/i  into  ax^X2a\     Prod. 

103.  It  will  be  easy  to  see  that  when  the  multiplier  and 
multiplicand  consist  of  any  quantity  repeated  as  a  factor,  this 
factor  will  be  repeated  in  the  product,  as  many  times  as  in 
the  multiplier  and  multiplicand  together. 
Mult.     ax«X«     Here  rt  is  repeated //iree  ^imes  as  a  factor. 
Into        ax«  Here  it  is  repeated  twice. 


Prod.     ax«X^X«X«-     Here  it  is  repeated ^lye  times. 

The  product  of  hhhh  into  hhh,  is  bbbbbbb. 
The  product  of  2a: X 3a; X 4a;  into  oxx^^t  is  2a;x3a;x4a;X 
5a;x6a;. 

104.  But  the  nwmemZ  co-e^dmf 5  of  several  fellow-factors 
may  be  brought  together  by  multiplication. 

Thus  2ax36  into  4ax56  is  2ax36x4«x5&,  or  \20aabb. 
For  the  co-efficients  a.Ye  factors,  (Art.  41.)  and  it  is  imma- 
terial in  what  order  these  are  arranged.      (Art.  95.)    So  that 
2ax36x4ax56=r2x3x4x5x«XttX^X^=120rta66. 
The  product  of  3ax46/i  into  dmx^y,  is  SSOabhmy. 
The  product  of  4^x6^^  into^2a;+l,  is  48bdx-{-24bd. 

105.  The  examples  in  multiplication  thus  far  have  been 
confined  to  positive  quantities.  It  will  now  be  necessary  to 
consider  in  what  manner  the  result  will  be  affected,  by  mul- 
tiplying positive  and  negative  quantities  together.     We  shall 

That  -|-  into  -{-produces  -\- 

-  into  -j-  - 
•4"  into  ~  - 

-  into  -  -f- 


MULTIPLICATION  37 

All  these  may  be  comprised  in  one  general  rule,  which  it 
will  be  important  to  have  ahvays  familiar.      If  the  signs  of 

THE  FACTORS  ARE  ALIKE,  THE  SIGN  OF  THE  PRODUCT  WILL 
BE  AFFIRMATIVE  ;  BUT  IF  THE_ SIGNS  OF  THE  FACTORS  ARE 
UNLIKE,  THE   SIGN  OF  THE   PRODUCT  wi_LirBE^EGATiVE. 

106.  Tlie  first  case,  that  of  -J-  into  -[-,  needs  no  farther 
illustration.  The  second  is  -  into  -j-,  that  is,  the  multipli- 
cand is  negative,  and  the  multiplier  positive.  Here  -a 
into  -(-4  is  -  4a.     For  tbe  repetitions  of  the  multiplicand  are, 

-a-a-a-az=-4a. 
Mult.       b~Sa  2a-m         h-U-A  a-2-7d-x 

Into       6j/  Sh-Lx       2y  36+/i 


Prod.     6by-lSay  2hy-6dy-Sij 


107.  In  the  two  preceding  cases,  the  affirmative  sign  pre- 
fixed to  the  multiplier  shows,  tliat  the  repetitions  of  the  mill-, 
tiplicand  are  to  be  added  to  the  other  quantities  with  which 
the  multiplier  is  connected.  But  in  the  two  remaining  cases, 
the  negcitive  sign  prefixed  to  the  multiplier,  indicates  that 
the  sum  of  the  repetitions  of  tire  multiplicand  are  to  be  sub- 
tracted from  the  other  quantities.  (Art.  90.)  And  this  sub- 
traction is  performed,  at  the  time  of  multipljdng,  by  making 
the  sign  of  the  product  opposite  to  that  of  the  multiplicand. 
Thus  -}-a  into  -4  is  -4a.  For  the  repetitions  of  the  multi- 
plicand are, 

-|-a-|-«+"+''^=4-4^- 

But  this  sum  is  to  be  subtracted,  from  the  other  quantities 
with  which  the  multiplier  is  connected.  It  will  then  become 
-4a.      (Art.  82.) 

Thus  in  the  expression  6 -(4x^5)  it  i^  manifest  that4x« 
is  to  be  subtracted  from  b.  Now  4X(^  is  4a,  that  is  -f-4a. 
But  to  subtract  this  from  b,  the  sign  -\-  must  be  changed 
into  -.  So  that  6  -  (4  x  «)  is  b-  4a.  And  a  x  -  4  is  there- 
fore -  4a. 

Again,  suppose  the  multiplicand  is  a,  and  the  multiplier 
(6-4.)  As  (6-4)  is  equal  to  2,  the  product*  will  be  equal 
to  2a.  This  is  less  than  the  product  of  6  into  a.  To  obtain 
then  the  product  of  the  compound  multiplier  (6-4)  into  a, 
we  must  subtract  the  product  of  the  negative  part,  fiom  thai 
of  the  positive  part. 


2S35G0 


38  ALGEBRA. 

Multiplying         a  >  .     ,  C  Multiplying  a 

Into  6-45    '^'^''''''^^' Unto  2 


And  the  product  6a -4a,  is  the  same  as  the  product  2a. 
Therefore  a  into  -  4,  is  -  4a. 

But  if  the  multipher  had  been  (6-\-4,)  the  two  products 
must  liave  been  added. 

Multiplying  a  )   •     ,  (  Multiplying  a 

Into  6-f  4  I  ^^  ^^^  ^^^^  ^^  I  Into  10 


And  the  prod.         6a-f-4a  is  the  same  as  the  product     10a. 

This  shows  at  once  the  difference  between  multiplying  by 
a  positive  factor,  and  multiplying  by  a  negative  one.  In  the 
former  case,  the  sum  of  the  repetitions  of  the  multiplicand  is 
to  be  added  to,  in  the  latter,  subtracted  from,  the  other  quan- 
tities, with  which  the  multiplier  is  connected.  For  every 
negative  quantity  must  be  supposed  to  have  a  reference  to 
some  other  which  is  positive ;  though  the  two  may  not 
always  stand  m  connection,  when  the  multiplication  is  to  be 
performed. 

Mult,     a-^b  3^?/+/i.T+2         Sh  +3 

Into        b-x  mr  ~ab  ad -6 


Prod,  ab+bb -ax-bx  ^adh-[-^ad -  I8/1  -  18 

108.  If  two  negatives  be  multiplied  together,  the  product 
will  be  affirmative  :  - 4x-«=+4a.  In  this  case,  as  in  the 
preceding,  the  repetitions  of  the  multiplicand  are  to  be  sub- 
tracted, because  the  multiplier  has  the  negative  sign.  These 
repetitions,  if  the  multiplicand  is  -  a,  and  the  multiplier  -  4, 
are -a-a-a-a=-4a.  But  this  is  to  be  subtracted  by 
changing  the  sign.     It  then  becomes  -\-Aa. 

Suppose -a  is  multiplied  into  (6-4.)  As  6-4=2,  the 
product  is,  evidently,  twice  the  multiplicand,  that  is,  -2a. 
But  if  we  multiply  -  a  into  6  and  4  separately ;  -  a  into  6 
is  -  6a,  and  -  a  into  4  is  -  4a.  (Art.  106.)  As  in  the  multi- 
plier, 4  is  to  be  subtracted  from  6  ;  so,  in  the  product,  -  4a 
must  be  subtracted  from  -  6a.  Now  -  4a  becomes  by  sub- 
traction -f-4a.  The  whole  product  then  is  -  6ft-|-4a  which  is 
equal  to  -  2a.  •  Or  thus, 

Muhiplying       -  a  >  .     ,  C  Multiplpng  -  a 

Into  6-45  ^^^^^  ^^^^^^  ^^  I  Into  2 

And  the  prod.     -  6a+4a,  is  equal  to  the  product     -  3«. 


MULTIPLICATION.  39 

It  is  often  considered  a  great  mystery,  that  the  product  of 
two  negatives  sliould  be  affirmative.  But  it  amoimts  to  no- 
thing more  than  this,  that  the  subtraction  of  a  negative  quan- 
tity, is  equivalent  to  the  addition  of  an  affirmative  one; 
(Art.  8L)  and,  therefore,  that  the  repeated  subtraction  of  a 
negative  quantity,  is  equivalent  to  a  repeated  addition  of  an 
affirmative  one.  *  Taking  off  from  a  man's  hands  a  debt  of 
ten  dollars  every  month,  is  adding  ten  dollars  a  month  to  the 
value  of  his  property. 

Mult.       a- 4  3d-hy-2x   Say-b 

Into       36-6  46-7  6^-1 


Prod.     Sub  -  126  -  6a+24  \8axy  -  6bx -  Say-\-b 

Multiply  Sad  -ah -7  into  4-dy-  hr. 
Multiply  2hy-\-Sm  -  1  into  4d-2x-{-S. 

1 09.  As  a  negative  multiplier  changes  the  sign  of  the  quan- 
tity which  it  multi})lies  ;  if  there  are  several  negative  factors 
to  be  multiplied  together. 

The  two  first  will  make  the  product  positive; 

The  third  will  make  it  negative; 

The  fourth  will  make  il  positive,  &c. 
Thus-ax-^  =+a^      1  p^co  factors. 

^abx-c=  -abc      I  ^^    (  three, 

-abcx -d=-\-abcd    (        ^  \  four, 

-\-abcdx-^=-a'bcdej  ifi^^^- 

That  is,  the  product  of  any  even  number  of  negative  fac- 
tors IS  positive;  but  the  product  of  any  odd  number  of  nega- 
tive factors  is  negative. 

Thus-ax-«  =  «<^  And -ax  -«X  -«X  -a=aaaa 

-ax -ax  -a=-aaa       -ax-«X-aX-«X  -  a=-  aaaaa 

The  product  of  several  factors  wliich  are  ail  positive,  is  in- 
variably positive. 

110.  Positive  and  negative  terms  may  frequently  balance 
each  other,  so  as  to  disappear  in  the  product.    (Art.  77.)     A 
star  is  sometimes  put  in  the  place  of  a  deficient  term. 
Mult,  a-b  mm-yy         aa-\-ab-\-bb 

Into     a-\-b  mm-{-yy  a-b 


aa-ab  aaa-\-aab-{-abb 

-\-ab  -bb  -  aab  -  abb  -  bbb 


Prod,  aa  *  -bb  aaa     *   *  _  666 


40  ALGEBRA, 

111.  For  many  purposes,  it  is  sufficient  merely  to  indicate 
the  multfplication  of  compound  quantities,  without  actually 
multiplying  the  several  terms.     Thus  the  product  of 
a-]-b-\-c  into  h-\-m-\-y,  is  (a-f-6+c)  x  {h-\-m-\-y.)     (Art.  40.) 
The  product  of 

a-\-m  into  h-\-x  and  d-\-y,  is  (a-\-m)  X  {h-{-x)  X  (^+2/-) 

By  this  method  of  representing  multiplication,  an  important 
advantage  is  often  gained,  in  preserving  the  factors  distinct 
from  each  other. 

When  the  several  terms  are  multiplied  in  form,  the  expres 
sion  is  said  to  be  expanded.     Thus, 

{a-\-b)  X  (c+(^)  becomes  when  expanded  ac-\-ad-\-bc-}-bd 

112.  With  a  given  multiphcand,  the  less  the  nuiltipher, 
the  less  will  be  the  product.  If  then  the  multiplier  be 
reduced  to  nothing,  the  product  will  be  nothing.  Thus  axO 
=  0.  And  if  0  be  one  of  a7iy  number  of  fellow-factors,  the 
product  of  the  whole  will  be  nothing. 

Thus,  abxcX^dxO^SabcdxO  =  0. 
And  {a-irb)x{c+d)x{h-m)xO  =  0. 

113.  Although,  for  the  sake  of  illustrating  the  different 
points  in  multiplication,  the  subject  has  been  drawn  out  into 
a  considerable  number  of  particulars;  yet  it  will  scarcely  be 
necessary  for  the  learner,  after  he  has  become  familiar  with 
the  examples,  to  burden  his  memory  with  any  thing  moie 
than  the  following  general  rule. 

Multiply  the  letters  and  co-efficients  of  each  term 

IN  THE  MULTIPLICAND,  into  THE  LETTERS  AND  CO-EFFICIENTS 
OF  EACH  TERM  IN  THE  MULTIPLIER;  AND  PREFIX  TO  EACH  TERM 
OF  THE  PRODUCT,  THE  SIGN  REQUIRED  BY  THE  PRINCIPLE,  THAT 
LIKE  SIGNS  PRODUCE-f-,  AND  DIFFERENT  SIGNS  -  . 

1 .  Mult.  a-\-Sb  -  2  into  4a  -  66  -  4. 

2.  Mult.  4abxxx2  mio  Smy~\-\-h. 

3.  Malt.  {7 ah  -  ^)  X  4  into  4x  X  3  X  5  X  ^. 

4.  Mult.  {6ab-hd^l)x2mto  {S-\-^x-\)xd. 

5.  Muh.  3fly+?/-4-f/iinto  (c/-[-;c)x(/t4-V-) 

6.  Mult.  6ax-{4h-d)  into  (6+l)x(/i+i0 

7.  Muh.  lay -l-\-hx{d-x)  into  -  {r-]-S - 4m.) 


DIVISION.  41 


SECTION  V. 


DIVISION. 


Art.  114.  IN  multiplication,  we  have  two  factors  given, 
and  are  required  to  find  their  product.  By  multiplying  the 
factors  4  and  6,  we  obtain  the  product  24.  But  it  is  fre- 
quently necessary  to  reverse  this  process.  The  number  24, 
and  one  of  the  factors  may  be  given,  to  enable  us  to  find  the 
other.  The  operation  by  which  this  is  eflfected,  is  called 
Division.  We  obtain  the  number  4,  by  dividing  24  by  6. 
The  quantity  to  be  chvided  is  called  the  dividend  ;  the  given 
factor,  tlie  divisor  ;  and  that  which  is  required,  the  quotient. 

115.  DIVISION  IS  FINDING  A  QUOTIENT,  WHICH  MULTI- 
PLIED INTO  THE  DIVISOR  WILL  PRODUCE   THE  DIVIDEND.* 

In  multiplication  the  multiplier  is  always  a  number.  (Art. 
91.)  And  the  product  is  a  quantity  of  the  same  kind,  as  the 
multiplicand.  (Art.  92.)  The  product  of  3  rods  into  4,  is  12 
rods.  When  we  come  to  division,  the  product  and  either  of 
the  factors  may  be  given,  to  find  the  other  :  that  is. 

The  divisor  may  be  a  number,  and  then  the  quotient  will 
be  a  quantity  of  the  same  kind  as  the  dividend  ;  or, 

The  divisor  may  be  a  quantity  of  the  same  kind  as  the 
dividend  ;  and  then  tlie  quotient  will  be  a  number. 

Thus  12  rods'i-4:=S  rods.     But  12  rods -i-S  rods  =  4. 

And  12  rodS'i-24  =  irod.     And  12  rods-^24  rods^^ 

In  the  first  case,  the  divisor  being  a  number,  shows  into 
how  many  parts  the  dividend  is  to  be  separated  ;  and  the  quo- 
tient sliows  what  tliese  parts  are. 

If  12  rods  be  divided  into  3  parts,  each  will  be  4  rods  long. 
And  if  12  rods  be  divided  into  24  parts,  each  will  be  half  a 
rod  long. 

In  the  other  case,  if  the  divisor  is  less  than  the  dividend, 
the  former  shows  into  ichat  paits  the  latter  is  to  be  divided  ; 
and  the  quotient  shows  how  many  of  these  parts  are  contained 


*  The  remainder  is  here  supposed  to  be  included  in  the  quotient,  as  is  com- 
monly the  case  in  algebra. 

r. 


42  ALGEBRA. 

in  the  dividend.     In  other  words,  division  in  this  case  con- 
sists in  finding  hoio  often  one  quantity  is  contained  in  another. 

A  line  of  3  rods,  is  contained  in  one  of  1 2  rods,  four  times. 

But  if  the  divisor  is  greater  than  the  dividend,  and  yet  a 
quantity  of  the  same  kind,  the  quotient  shows  what  part  of 
the  divisor  is  equal  to  the  dividend. 

Thus  one  half  of  24  rods  is  equal  to  12  rods. 

116.  As  the  product  of  the  divisor  and  quotient  is  equal  to 
the  dividend,  the  quotient  may  be  found,  by  resolving  the 
dividend  into  two  such  factors,  that  one  of  them  shall  be  the 
divisor.     The  other  will,  of  course,  be  the  quotient. 

Suppose  abd  is  to  be  divided  by  a.  The  factor  a  and  bd 
will  produce  the  dividend.  The  first  of  these,  being  a  divi- 
sor, may  be  set  aside.     The  other  is  the  quotient.     Hence, 

When  the  divisor  is  found  as  a  factor  in  the  divi- 
dend,'' THE     DIVISION     IS    PERFORMED    BY    CANCELLING    THIS 


abxy 
ax 


FACTOR. 

Divide    ex 

dh 

drx 

hrny 

dhxy 

abed 

By          c 

d 

dr 

km 

dy 

b 

Quot.      x  X  hx  by 


In  each  of  these  examples,  the  letters  which  are  common 
to  the  divisor  and  dividend,  are  set  aside,  and  the  other  let- 
ters form  the  quotient.  It  will  be  seen  at  once,  that  the  pro- 
duct of  the  quotient  and  divisor  is  equal  to  the  dividend. 

117.  If  a  letter  is  repeated  in  the  dividend,  care  must  be 
taken  that  the  factor  rejected  be  only  equal  to  the  divisor. 

Div.    aab        bbx        aadddx        aammyy        aaaxxxh        yyy 
By       a  b  ad  amy  aaxx  yy 


Quot.  ab  addx  ahx 


In  such  instances,  it  is  obvious  that  we  are  not  to  reject 
every  letter  in  the  dividend  which  is  the  same  with  one  in  the 
divisor. 

118.  If  the  dividend  consists  of  any  factors  whatever,  ex- 
punguig  one  of  them  is  dividing  by  it. 


DIVISION.  43 

Div.     a  {b+d)  a  (bJrd)      (bJ^x){c-\-d)        (b-{.y)x{d-h)x 
By       a  b-^d  b-]-x  d-h 


Quot.  b^d  a  c^d  (^bJ^y)xx 


In  all  these  instances  the  product  of  the  quotient  and  divi- 
sor is  equal  to  the  dividend  by  Art.  111. 

119.  In  performing  multiphcation,  if  the  factors  contain 
numeral  figures,  these  are  multiplied  into  each  other.  (Art. 
96.)  Thus  3a  into  76  is  2lab.  Now  if  this  process  is  to  be 
reversed,  it  is  evident  that  divichng  the  number  in  the  product, 
by  the  number  in  one  of  the  factors,  will  give  the  number  in 
the  other  factor.     The  quotient  of  21  ab-^Sa  is  76.     Hence, 

In  division,  if  there  are  numeral  co-efficients  prefixed  to  the 
letters,  the  co-efficient  of  the  dividend  must  be  divided,  by  the  co- 
efficient of  the  divisor. 

Div.  6ab         Udxy         2odhr         12xy         S4drx         20hm 
By     26  4dx  dh  6  S4  m 


Quot.3a  25r  drx 


120.  When  a  simple  factor  is  multiphed  into  a  compound 
one,  the  former  enters  into  every  term  of  ihe  latter.       (Art. 
98.)     Thus  a  into  b-\-d,  is  ab-\-ad.     Such  a  product  is  easily 
resolved  again  into  its  original  factors. 
Th-Ms  ab-{-ad=ax{b-{-d). 

ab-\-ac-\-ah=ax{b-T-c-\-h), 

amh-\-amx-\-amy=zam  X  {h-\-x-\-y). 

4ad-\-Sah-\-\2amJ^4ay  =  4ax(d4-^h-\-Sm-\-y), 
Now  if  the  whole  quantity  be  divided  by  one  of  these  factors, 
accordhig  to  Art.  118,  the  quotient  will  be  the  other  factor. 

Thus,   (ab-\-ad)-^a=b-\-d.     And   {ab-{-ad)-^{b-\-d)=a. 
Hence, 

If  the  divisor  is  contained  in  every  term  of  a  compound  divi- 
dend, it  must  be  cancelled  in  each. 

Div.     ab-\-ac  bdh-\-bdy         aah-\-ay         drx-{-dhx-\-dxy 

By       a  bd  a  dx 


Quot.  6-f-c       ,  «^+!/ 


And  if  there  are  co-efficients,  these  must  be  divided,  in  each 
tenia  also. 

4^' 


44  ALGEBRA. 

Div.       6ab+\2ac      lOdry-\-l6d      12/ia;+8       35c?m+14c?a; 
By         3a  2d  4  7d 


Quot.    264-4C  Shx+2 


TT-yi 


121.  On  the  other  hand,  if  a  compound  expression  contain- 
ing any  factor  in  every  term,  he  divided  by  the  other  quantities 
connected  by  their  signs ,  the  quotient  will  be  that  factor.  See  the 
first  part  of  the  precedmg  article. 

Div.       ab-\-ac-\-ah    amh-\-amx-\-amy    4ab-\-8ay    ahm-\-chy 
By  b-\-c4-h         h-]-x-{-y  b-\-2y         m+y 


Quot.     a  4a 


122.  In  division,  as  well  as  in  multiplication,  the  caution 
must  be  observed,  not  to  confound  terms  with  factors.  See 
Art.  99. 

Thus{ab+ac)-^a=b^c.     (Art.  120.) 
But    labX(i'C)-T-a=aabc-^a=abc. 
And  {ab-\-ac)-7-{b-]-c)=a.     (Art.  121.) 
But    labxci'C)-T-lbX(^)=ct'Cibc-i-bc=aa. 

123.  In  division,  the  same  rule  is  to  be  observed 
respecting  the  signs,  as  in  multiplication  ;  that  is, 
if  the  divisor  and  dividend  are  both  positive,  or 
both  negative,  the  quotient  must  be  positive  :  if 
one    is    positive   and  the  other  negative,  the  quo-^ 

TIENT    MUST    BE    NEGATIVE.        (Art.   105.) 

This  is  manifest  from  the  consideration  that  the  product  of 
the  divisor  and  quotient  must  be  the  same  as  the  dividend. 

If  ^ax+b  =+ab\  r +a6-^+6  =+a 

7«X+f=-«Mthen  \   -'it^l^l'' 
'J^aX-b=-abL  i   -  ab-^- b=z-{-a 

^aX-b  =  -\-abj  \ -\-ab-i--b=  -  a 

Div.      abx         Sa-lOay      Sax-6ay      Qamxdh 
By         -a  -2a  3a  -2a 


Quot.     -bx       -  4-1- 5y  -Smxdh=  -  Shdm 

— ^. 


DIVISION.  45 

124.  If  the  letters  of  the  divisor  are  not  to  be  found 
in  the  dividend,  the  division  is  expressed  by  writing 

the    divisor    UNDER  THE  DIVIDEND,  IN  THE  FORM  OF  A  VUL- 
GAR FRACTION. 

xy  d-x 

Thus  xy-i-a=  — ;  and  (c?  -  x)  -f-  -  h=  ——r 

This  is  a  method  of  denoting  division,  rather  than  an  actual 
peifonning"  of  the  operation.  But  the  purposes  of  division 
may  frequently  be  answered,  by  these  fractional  expressions. 
As  they  are  of  the  same  nature  wilh  other  vulgar  fractions, 
they  may  be  added,  subtracted,  multiplied,  &c.  See  the 
next  section. 

125.  When  the  dividend  is  a  compound  quantity,  the  divi- 
sor may  either  be  placed  under  the  lohole  dividend,  as  in  the 
preceding  instances,  or  it  may  be  repeated  under  each  temiy 
taken  separately.  There  are  occasions  when  it  will  be  con- 
venient to  exchange  one  of  these  forms  of  expression  for  the 
other. 

h—\—c         h     c 
Thus  6-j-c  divided  by  x,  is  either ,  or    -| — . 

X  X       X 

a-\-b 
And  a-\-b  divided  by  2,  is  either  ■"^~~,  that  is,  half  the  sum 

of  a  and  b;  or— -|--,  that  is,  the  sum  of  half  a  and  half  b. 

For  it  is  evident  that  half  the  sum  of  two  or  more  quantities, 
is  equal  to  the  sum  of  their  halves.  And  the  same  principle 
is  applicable  to  a  thiid,  fourth,  fifth,  or  any  other  portion  of 
the  dividend. 

So  also  a-b  divided  by  2,  is  either     ~   .,  or  — 

•^    '  2    '        2     2 

For  half  the  difference  of  two  quantities  is  equal  to  the  dif- 
ference of  their  halves. 

a-2b-\-h     a     2b     h  Sa-c       Sa         c 

^°        m        =m-m+m    ^^^^1^=— -— ' 

126.  If  some  of  the  letters  in  the  divisor  are  in  each  term 
of  the  dividend,  the  fractional  expression  may  be  rendered 
more  simple,  by  rejecting  equal  factors  from  the  numerator 
and  denominator. 


46 

Div.    ab 

By      ac 

b 
c 

dhx 
dy 

ALGEBRA. 

ahm  -  Say 
ab 

ab+bx 
by 

2am 
2xy 

ab 
Quot.  —  01 
ac 

hin  -  Sy 
b 

am 
xy 

These  reductions  are  made  upon  the  principle,  that  a  given 
divisor  is  contained  in  a  given  dividend,  just  as  many  times, 
as  double  the  divisor  in  double  the  dividend ;  triple  the  divi- 
sor in  triple  the  dividend,  &c.    See  the  reduction  of  fractions. 

127.  If  the  divisor  is  in  some  of  the  terms  of  the  dividend, 
but  not  in  all ;  those  which  contain  the  divisor  may  be  divi- 
ded as  in  Art.  116,  and  the  others  set  down  in  the  form  of  a 
fraction. 

Thus  (ab-\-d)-^a  is  either  — ^^^.   or  — -1 — ,  oxb-\ — . 
^      '    ^  a  a^  a         ^  a 

-Div.    dxy-]-rx-hd         2ah-\-ad-{~x  bm-\-Sy       27ny-\-dh 

By      a;  a  -b  2m 


^noi.  dy+r-'^  _,„^ -^ 


128.  The  quotient  of  any  quantity  divided  by  itself  or  its 
equal,  is  obviously  a  unit. 

Thus-=:1.  And^-  =  1.  And-— —  =  1.  And-f-T — ^,  =1. 
a  Sax  4-|-2  a-\-b-Sh 

Div.    ax-\-x         Sbd-Sd         4axy  -  4a-{-8ad       Sab-\-S-6m 
By      X  Sd  4a  3 


Quot.  a+1  xy-l+2d 


Cor.  If  the  dividend  is  greater  than  the  divisor,  the  quo- 
tient must  be  greater  than  a  unit :  But  if  the  dividend  is  less 
than  the  divisor,  the  quotient  must  be  less  than  a  unit. 


DIVISION.  .47 


PROMISCUOUS    EXAMPLES. 

1.  Divide  12aby-\r^abx  -  lSbbm-\-24b,  by  6b. 

2.  Divide  16a- 12+8^+4 -20af^x+m,  by  4. 

3.  Divide  (a-2/i)  x(3m+?/)  x^^,  by  {a  -  2h)  x  {^rn+y) 

4.  Divide  ahd  -  4ad-\-Say  -  a,  by  hd  -  4d-\-Sy  -  1. 

5.  Divide  ax  -  ry-{-ad  -  imy  -6-\-a,  by  -  a. 

6.  Divide  amy-\-Smy  -  mxy-\-am  -  rf,  by  -  dmy. 

7.  Divide  ard  -  6a+2r  -  hd-\-Q,  by  2ar(?. 

8.  Divide  Qax  -  S-\-2xy-\-4  -  6hy,  by  4axy. 

129.  From  the  nature  of  division  it  is  evident,  that  the 
value  of  the  quotient  depends  both  on  the  diWsor  and  the 
dividend.  With  a  given  divisor,  the  greater  the  dividend, 
the  greater  the  quotient.  And  with  a  given  dividend,  the 
greater  the  divisor,  the  less  the  quotient.  In  several  of  the 
succeeding  parts  of  algebra,  particularly  the  subjects  of  frac- 
tions, ratios,  and  proportion,  it  will  be  unportant  to  be  able 
to  determine  what  change  will  be  produced  in  the  quotient, 
by  increasing  or  diminishing  either  tlie  divisor  or  the  dividend. 

If  the  given  dividend  be  24,  and  the  divisor  6  ;  the  quotient 
will  be  4.  But  this  same  dividend  may  be  suppose|i  to  be 
multiplied  or  divided  by  some  other  number,  beforekit  is 
divided  by  6.  Or  the  divisor  may  l)e  multiplied  or  divided 
by  some  other  number,  before  it  is  used  in  dividing  24.'  '^n 
each  of  these  cases,  the  quotient  will  be  altered.  '-^^., 

130.  In  the  first  place,  if  the  given  divisor  is  contained  in 
the  given  dividend  a  certain  number  of  times,  it  is  obvious 
that  the  same  divisor  is  contained, 

In  double  that  di\ndend,  twice  as  many  times  ; 

In  triple  the  dividend,  thrice  as  many  times,  &c. 

That  is,  if  the  divisor  remains  the  same,  multiplying  the 
dividend  by  any  quantity,  is,  in  effect,  multiplying  the  quotient 
by  that  quantity. 

Thus,  if  the  constant  divisor  is  6,  then  24-j-6  =  4  the 
quotient. 

Multiplying  the  dividend  by  2,  2  X  24—6  =  2x4 

Muhi plying  by  any  number  n,  nx 24-7-6=^x4 


48  ALGEBRA. 

131.  Secondly,  if  tlie  given  divisor  is  contained  in  the 
given  di\ddend  a  certain  number  of  times,  the  same  divisor 
is  contained, 

In  half  that  dividend,  half  as  many  times  ; 

In  one  third  of  the  dividend,  one  third  as  man)/  times,  &c. 

That  is,  if  the  divisor  remains  the  same,  dividing  the  divi- 
dend by  any  other  quantity,  is,  in  effect,  dividing  the  quotient 
by  that  quantit}^ 

Thus  24-i-6=4 

Dividing  the  dividend  by  2,  ^24-7-6  =  J  4 

Dividing  by  ?i,  ^24-7-6=-^ 

132.  Thirdly,  if  the  given  divisor  is  contained  in  the  given 
dividend  a  certain  number  of  times,  then,  in  the  same  divi- 
dend. 

Twice  that  divisor  is  contained  only  half  as  many  times ; 
Three  times  the  divisor  is  contained  one  third  as  many  times. 

That  is,  if  the  dividend  remains  the  same,  multiplying  the 
divisor  by  any  quantity,  is,  in  effect,  dividing  the  quotient  by 
that  quantit}'-. 

Thus       '  24—6  =  4 

Multiplying  the  divisor  by  2,  24-r2x6=:| 

Multiplying  by  n,  24-j-n  X  6  =  i 

133.  Lastly,  if  the  given  divisor  is  contained  in  the  given 
dividend  a  certain  number  of  tunes,  then,  in  the  same  divi- 
dend, 

Half  that  divisor  is  contained  ticice  as  many  times  ; 

One  third  of  the  divisor  is  contained  thrice  as  many  times. 

That  is,  if  the  dividend  remains  the  same,  dividing  the  divi- 
sor by  any  other  quantity,  is,  in  effect,  multiplying  the  quotient 
by  that  quantity. 

Thus  24-^6=:4 

Dividing  the  divisor  by  2,  24-f-i6=:2x4 

Dividing  by  n,  24-^^6 =nX  4 

For  the  method  of  performing  division,  when  the  divisor 
and  dividend  are  both  compound  quantities,  see  one  of  the  fol- 
lowing sections. 


FRACTIONS.  49 


SECTION  V. 


FRACTIONS.* 

Art.  134.  EXPRESSIONS  in  the  form  of  fractions  occur 
more  frequently  in  Algebra  than  in  arithmetic.  Most  in- 
stances in  division  belong  to  this  class.  Indeed  the  numera- 
tor of  every  fraction  may  be  considered  as  a  dividend,  of 
which  the  denominator  is  a  divisor. 

According  to  the  common  definition  in  arithmetic,  the 
denominator  shows  into  what  parts  an  integral  unit  is  sup- 
posed to  be  divided  ;  and  the  numerator  shows  how  many 
of  these  parts  belong  to  the  fraction.  But  it  makes  no  dif- 
ference, whether  the  luhole  of  tlie  numerator  is  divided  by 
the  denominator  ;  or  only  one  of  the  integral  units  is  divided, 
and  then  the  quotient  taken  as  many  times  as  the  number  of 
units  in  the  numerator.  Thus  |  is  the  same  as  |-j-|-]-J. 
A  fourth  part  of  three  dollars,  is  equal  to  three  fourths  of  one 
dollar. 

135.  The  vcdue  of  a  fraction,  is  the  quotient  of  the  nume- 
rator divided  by  the  denominator. 

Thus  the  value  of  -  is  3.     The  value  of  —  is  a. 
2  6 

From  this  it  is  evident,  that  whatever  changes  are  made 
in  the  terms  of  a  fraction  ;  if  the  quotient  is  not  altered,  the 
value  remains  the  same.  For  any  fraction,  therefore,  we 
may  substitute  any  other  fraction  which  will  give  the  same 
quotient. 

rr.,4     10     4ba     Sdrx     6+2  .         t^      ,i.  ,•     ,  • 

lnus-= — = —  = —  _J — ,&c.     hoi  the  quotient  m 

2      5      2ba     Urx     3-fl  ^ 

each  of  these  instances  is  2.  <, 

136.  As  the  value  of  a  fraction  is  the  quotient  of  the  nu- 
merator divided  by  the  denominator,  it  is  evident  from  Art. 
128,  that  when  the  numerator  is  equal  to  the  denominator,  the 
value  of  the  fraction  is  a  unit ;    when  the  numerator  is  less 

*  Horsley's  Mathematics,  Camus'  Arithmetic,  Emerson,  Euler,  Saundersan, 
and  Ludlam. 

7 


50  ALGEBRA. 

than  the  denominator,  the  value  is  less  than  a  unit;  and  when 
the  numerator  is  greater  than  the  denominator,  the  value  is 
greater  than  a  unit. 

The  calculations  in  fractions  depend  on  a  few  general 
principles,  which  will  here  be  stated  in  connexion  with  each 
other. 

137.  If  the  denominator  of  a  fraction  remains  the  same,  mul- 
tiplying the  NUxMERATOR  by  any  quantity,  is  multiplying  the 
VALUE  by  that  quantity  ;  and  dividing  the  numerator,  is  dividing 
the  value.  For  the  numerator  and  denominator  are  a  divi- 
dend and  divisor,  of  which  the  value  of  the  fraction  is  the 
quotient.  And  by  Art.  130  and  131,  multiplying  the  divi- 
dend is  in  effect  multipl3nng  the  quotient,  and  dividing  the 
dividend  is  dividing  the  quotient. 

mi        •     xi       r      X-         (lb       3a6      7abd     \ab   ^ 

Thus  m  the   fractions  — ,    ,    ,    5 — ,  &c. 

a  a  a  a 

The  quotients  or  values  are  b,       Sb,       7bd,     ^b,  &c. 

Here  it  Avill  be  seen  that,  while  the  denominator  is  not 

altered,  the  value  of  the  fraction  is  multiplied  or  divided  by 

the  same  quantity  as  the  numerator. 

Cor.  With  a  given  denominator,  the  greater  the  mune- 
rator,  the  greater  will  be  the  value  of  the  fraction  ;  and,  on  the 
other  hand,  the  greater  the  value,  the  greater  the  numerator, 

138.  If  the  numerator  remains  the  same,  multiplying  the  de- 
nominator by  any  quantity,  is  dividing  the  value  by  that  quantity  ; 
and  dividing  the  denominator,  is  multiplying  the  value.  For 
multiplying  the  divisor  is  dividing  the  quotient ;  and  dividing 
the  divisor  is  multiplying  the  quotient.      (Art.  132,  133.) 

T    ,,      .      ,.         24ab     24ab     24ab     24ab    . 

In  the  fractions    — — ,    ,    ,     — —,  ccc. 

66  '    126         36  6 

The  values  are     4a,       2a,        8a,        24a,  &c. 

Cor.  With  a  given  numerator,  the  greater  the  denominator, 
the  less  will  be  the  value  of  the  fraction  ;  and  the  less  the 
value,  the  greater  the  denominator. 

139.  From  the  last  two  articles  it  follows,  that  dividing  the 
numerator  by  any  quantity,  will  have  the  same  effect  on  the 
value  of  the  fraction,  as  multiplying  the  denominator  by  that 
quantity  ;  and  multiplying  the  numerator  will  have  the  same 
effect,  as  dividing  the  denominator. 


FRACTIONS.  51 

140.  It  is  also  evident  from  the  preceding  articles,  that  if 

THE  ^UMERATOR  AND  DENOMINATOR  BE  BOTH  MULTIPLIED, 
OR  BOTH  DIVIDED,  BY  THE  SAME  QUANTITY,  THE  VALUE  OF 
THE    FRACTION    WILL    NOT    BE    ALTERED. 

bx     ahx     ohx     \hx     iabx      .  -r.      •  l.     ^ 

T»->^    6  =Tb  =  36  =lb=l^  '  ^'-      ^'"'  '"  '''^  °' 

these  instances  the  quotient  is  x. 

141.  Any  integral  quantity  may,  without  altering  its  value, 
be  thrown  into  the  form  of  a  fraction,  by  multiplying  the 
quantity  into  the  proposed  denominator,  and  taking  the  pro- 
duct for  a  numerator. 

^,  a     ah     ad  4- ah     6adh     ^  ^        ,  .     ^ 

Thus  a=j=-^  =-^^  =-g^,  &c.     For   the  quotient 

of  each  of  these  is  a. 

^     ,     ,     dx-{-hx        ^     ,       ,  ,      2drr-\-2dr 
So  dJrh=—^- .      And  r  +  1  =       ^j^      . 

142.  There  is  nothing,  perhaps,  in  the  calculation  of  alge- 
braic fractions,  which  occasions  more  perplexity  to  a  learner, 
than  the  positive  and  negative  signs.  The  changes  in  these 
are  so  frequent,  that  it  is  necessary  to  become  familiar  with 
the  principles  on  which  they  are  made.  The  use  of  the  sign 
which  is  prefixed  to  the  dividing  line,  is  to  show^  whether  the 
value  of  the  ichole  fraction  is  to  be  added  to,  or  subtracted 
from,  the  other  quantities  with  which  it  is  connected.  (Art. 
43.)  Tliis  sign,  therefore,  has  an  influence  on  the  several 
terms  taken  collectively.  But  in  the  numerator  and  de- 
nominator, each  sign  affects  only  the  single  term  to  which  it 
is  applied. 

The  value  of  ^  is  a.     (Art.  135.)     But  this  will  become 

negative,  if  the  sign  -  be  prefixed  to  the  fraction. 

ab  ^  ab 

Thus  7/+-^  =i/+a.    Bnty--^=y-a. 

So  that  changing  the  sign  which  is  before  the  whole  frac- 
tion, has  the  effect  of  changing  the  value  from  positive  to 
negative,  or  from  negative  to  positive. 

Next,  suppose  the  sign  or  signs  of  the  numerator  to  be 
changed. 

ab  ^      -ab 

By  Art.  123,-j  =+a.      But  —  =  -a. 


62  AXGEBRA. 

.     .ab-bc      ,               _,       -ab-\-hc 
And — i — =4-a-c.     But  r-^ — =~a-\-c. 

That  is,  by  changing  all  the  signs  of  the  numerator,  the 
value  of  the  fraction  is  changed  from  positive  to  negative,  or 
the  contrary. 

Again,  suppose  the  sign  of  the  denominator  to  be  changed. 

As  before  -r  =4-a.  But  ——=-a. 

0  —b 

143.    We   have  then,  this  general  proposition;    If    the 

SIGN  PREFIXED  TO  A  FRACTION,  OR  ALL  THE  SIGNS  OF  THE 
NUMERATOR,  OR  ALL  THE  SIGNS  OF  THE  DENOMINATOR  BE 
CHANGED  ;  THE  VALUE  OF  THE  FRACTION  WILL  BE  CHANGED, 
FROM  POSITIVE  TO  NEGATIVE,  OR  FROM  NEGATIVE  TO  POSI- 
TIVE. 

From  this  is  derived  another  important  principle.  As  each 
of  the  changes  mentioned  here  is  from  positive  to  negative, 
or  the  contrary  ;  if  any  two  of  them  be  made  at  the  same 
time,  they  will  balance  each  other. 

Thus  by  changing  the  sign  of  the  numerator, 

ab       .      I  -ab 

—  =4-a  becomes =  -a. 

b      ^  b 

But,  by  changing  both  the  numerator  and  denominator,  it 

becomes ^I — =-\-a,  Avhere  the  positive  value  is  restored. 
—  b 

By  changing  the  sign  before  the  fraction, 
2/-|-_  =y-\-a  becomes  y-—  =y-a. 

But  by  changing  the  sign  of  the  numerator  also,  it  becomes 
y where  the  quotient  -  a  is  to  be  subtracted  from   y, 

or  which  is  the  same  thing,  (Art.  81,)  -|- a  is  to  be  added, 
making  y-{-a  as  at  first.     Hence, 

144.  If  all  the  signs  both  of  the  numerator  and 
denominator,  or  the  signs  of  one  of  these  with  the 
sign  prefixed  to  the  whole  fraction,  be  changed  at 
the  saxme  time,  the  value  of  the  fraction  will  not  be 

ALTERED. 


FRACTIONS.  53 

6-6-6  6 

Thus  ^=3^= --2-= --2  =+3. 

6-6         6  -6 

Hence  the  quotient  m  division  may  be  set  dowTi  4n  different 

a      —c         a      c 
ways.     Thus  (a  -  c)  -^b,  is  either  -7  +  — ,   or  -  -  -. 

The  latter  method  is  the  most  common.  See  the  exam- 
ples in  Art.  127. 

REDUCTION  OF  FRACTIONS. 

145.  From  the  principles  which  have  been  stated,  are  de- 
rived the  rules  for  the  reduction  of  fractions,  wiiich  are  sub- 
stantially the  same  in  algebra,  as  in  arithmetic. 

A  FRACTION  MAY  BE  REDUCED  TO  LOWER  TERMS,  BY  DIVI- 
DING BOTH  THE  NUMERATOR  AND  DENOMINATOR,  BY  ANY  QUAN- 
TITY WHICH  WILL  DIVIDE  THEM  WITHOUT  A  REMAINDER. 

According  to  Art.  140,  this  will  not  alter  the  value  of  the 
fraction. 

ab     a  6clm     Sm  7m      1 

Thus  —r=-'    And -TTT- =  1—    And-.^ — =  — 
CO      c  ody      4y  Imr      r 

In  the  last  example,  both  parts  of  the  fraction  are  divided 
by  the  numerator. 

.     a-\-bc  1     ,       ^     ^     ^  am-\-ay      a 

Again,.     ,  ,  .  — -=-(Art.  118.)  And  ,      I  /  — 7- 
*      '{a+6c)xwi     m^  '  bm-\-by      b 

If  a  letter  is  in  every  term,  both  of  the  numerator  and  de 
nominator,  it  may  be  cancelled,  for  this  is  dividiiig  by  that 
letter,     (Art.  120.) 

Thus  ^^^^^+«y     3m+y     dry-\-dy  ^r-f-1 
■'     ad-\-ah~   d-\-h      dhy-dy      h-l 

If  the  numerator  and  denominator  be  divided  by  the  great- 
est common  measure,  it  is  evident  that  the  fraction  Avill  be 
reduced  to  the  lowest  terms.  For  the  method  of  finding  the 
greatest  corrmion  measure,  see  Sec.  xvi. 

146.  Fractions  of  different  denominators  may  be  re- 
duced TO  A  common    denominator,  BY  MULTIPLYING    EACH 


54  ALGEBRA. 


FOR  A  NEW  NUMERATOR  ;  AND  ALL  THE  DENOMINATORS  TO- 
GETHER, FOR  A  COMMON  DENOMINATOR. 

Ex.  1.  Reduce  — ,  and  -,  and—   to  a  common denominatcr. 
b  d  y 

axdxy=ady  ^ 

cXbXy=cby  >    the  three  nmneraiors. 

mxbxd=mbd  ) 

bXfiXy  =  bdy  the  common  denominator. 

The  fractions  reduced  are  f  /^,  and  _^,  and  -— — 

bdy  bdy  bdy 

Here  it  will  he  seen,  that  the  reduction  consists  in  muki- 
plying  the  numerator  and  denominator  of  each  fraction,  into 
all  the  other  denominators.  This  does  not  alter  the  value. 
(Art.  140.) 

"2.  Reduce  —,  and  —,  and  —, 
3?n  g  y 


3.  Reduce  — ,  and  _,  and     '     . 

3  X  d-\-h 

1  ^ 

4.  Reduce -,   and 


a-]-b  a  -b 

After  the  fractions  have  been  reduced  to  a  common  de- 
nominator, they  may  be  brought  to  lower  terms,  by  the  rule  in 
the  last  article,  if  there  is  any  quantity  which  will  divide  the 
denominator,  and  all  the  numerators  without  a  remainder. 

An  integer  and  a  fraction,  are  easily  reduced  to  a  common 
denominator.     (Art.  141.) 

Thus  a  and  -  are  equal  to-  and  — ,  or  —  and  — . 
c  \  c  c  c 

A.nia,b,±,    1  are  equal  to  "J!^,    ''JUV,   K    *? 
m      y  my       my      my     my 

147.  To  REDUCE  AN  IMPROPER  FRACTION  TO  A  MIXED 
QUANTITY,  DIVIDE    THE   NUMERATOR    BY   THE    DENOMINATOR, 

as  in  Art.  127. 

rriu       ab-\-bm-\-d  ,       ,  d 

Thus      ~     ~     z=:a-\-m-\--, 
b  b 


FRACTIONS.  55 

_    ,         am-a-\-ady-hr  . 

Kediice    ,  to  a  mixed  quantity. 

For  the  reduction  of  a  mixed  quantity  to  an  improper  frac- 
tion, see  Art.  150.  And  for  the  reduction  of  a  compound 
fraction  to  a  simple  one,  see  Art.  160. 

ADDITION  OF  FRACTIONS. 

148.  In  adding  fractions,  we  may  either  write  them  one 
after  the  other,  with  their  signs,  as  in  the  addition  of  integers, 
or  we  may  incorporate  them  into  a  single  fraction,  by  the  fol- 
lowing rule  : 

Reduce  the  fractions  to  a  common  denominator, 
make  the  signs  before  them  all  positive,  and  then  add 
their  numerators. 

The  common  denominator  shows  into  what  parts  the  inte- 
gral unit  is  supposed  to  be  divided  ;  and  the  numerators  show 
the  number  of  these  parts  belonging  to  each  of  the  fractions 
(Art.  134.)  Therefore  the  numerators  taken  together  sYi09 
the  whole  number  of  parts  in  all  the  fractions. 
2      1      1  3      111 

Thus,  ;^  =  ;^  +  y.      And-  =  y+y  +  ^. 

Therefore,  ?.+^=i+i+l4-* +1=^. 
7^7      7  ^7  ^7  ^7  ^7      7 

The  numerators  are  added,  according  to  the  rules  for  the 
addition  of  integers.  (Art.  69,  &c.)  It  is  obvious  that  the 
sum  is  to  be  placed  over  the  common  denominator.  To 
avoid  the  perplexity  whicli  might  be  occasioned  by  the  signs, 
it  will  be  expedient  to  make  those  prefixed  to  the  fractions 
uniformly  positive.  But  in  doing  this,  care  must  be  taken 
not  to  alter  the  value.  This  A\dll  be  preserved,  if  all  the  signs 
in  the  numerator  are  changed  at  the  same  time  with  that  be- 
fore the  fraction.     (Art.  144.) 

Ex.  1.  Add — and —  of  a  pound.    Ans.  llt-or 

16         16         ^  16        16 

It  is  as  evident  that  7%,  and  ^s  of  a  pound,  are  y"^  of  a 
pound,  as  that  2  ounces  and  4  ounces,  are  6  ounces. 

2.  Add  —  and  — .     First  reduce  them  to  a  common  denomi 
h  d 

nator.     They  will  then  be^  and  — ,  and  their  sum^— lt_f . 

bd  bd  bd 


56  ALGEBRA. 

3.  Given  ^  and  -  ?!!±^,  to  find  thek  sum. 

Ans.^and  -  ?!zM^3Am  ^^^  _  2dr±M^SJm^r^ 
d  3/i        ^dh  Mil  Sdh 

A  a       1  _h -m_a   .~~h-\-m_ay-hd-\-dm 
"d  yd  y       ~  dy         ' 

K  ^       A     d    _  -am.     dy    _  -  am-\-dy    ^am-dy 
y  —  m      —  my     —  my  —  my  my 

n      ct  lb     _aa-ab-\-ah-\-bb  _aa-\-bb        ,.       ^^  v 

a-}-6  a-b     aa-\-ab  - ab  -  bb     aa-bb' 

7.  Add  — to-Zl.    8.  Add  lito-^-.  Ans.  -6. 
d       m-r  2       7-3 

149.  For  many  purposes,  it  is  sufficient  to  add  fractions  in 
the  same  manner  as  integers  are  added,  by  writing  them  one 
after  another  with  their  signs.     (Art.  69.) 

Thus  the  sum  of—  and  —  and  -  — ,is  — +—  -  — - 
0  y  2m       b     y      2m 

In  the  same  manner,  fractions  and  integers  may  be  added. 

The  sum  of  a  and  —  and  3m  and  -  — ,  is  a-\-Sm-\—  -  — 
y  r  y      r 

150.  Or  the  integer  may  be  incorporated  with  the  fraction, 
by  converting  the  former  into  a  fraction,  and  then  adding  the 
numerators.     See  Art.  141. 

^,  .         J  b    .    a      b      am     b      am-\-b 

1  he  sum  oi  a  and  — ,  is  -7  4—  =  —  4—  = ■ — 

m        I'm      m    ' m        m 

^.  .  ^  ,       ,  h4-d  .    Sdm  -  Sdy4-h-{-d 

The  sum  of  3  J  and  -^^^,is o:__I_. 

m  -y  m-y 

Incorporating  an  integer  with  a  fraction,  is  the  same  as  re- 
ducing a  mixed  quantity  to  an  improper  fraction.  For  a  mixed 
quantity  is  an  integer  and  a  fraction.  In  arithmetic,  these 
are  generally  placed  together,  without  any  sign  between 


FRACTIONS.  57 

them.     But  ill  algebra,  they  are  distinct  terms.     Thus  2^  is 
2  and  ^,  which  is  the  same  as  2-|-i. 

Ex.   1.  Reduce  a-\-~  to  an  improper  fraction.     Ans.  — —-. 
b  0 

2.  Reduce  m+d—*:--,       Ans.   hm -  dm+dh~ dd-r_ 

h  —  d  h  —  d 

3.  Reduce    1  +  ^.     Ans.  -"t^.      4.  Reduce  1-A 

oh  m 

5.  Reduce  t+_L-.     6.  Reduce  3+::fl-^. 
d-y  3a 

SUBTRACTION  OF  FRACTIONS. 


51.  The  methods  of  performing  subtraction  in  algebra, 
depend  on  the  principle,  that  adding  a  negative  quantity  is 
equivalent  to  subtracting  a  positive  one  ;  and  v.  v.  (Art.  81.) 
For  the  subtraction  of  fractions,  then,  we  have  the  following 
simple  rule.     Change  the  fraction  to  be  I^ubtracted, 

FROM    POSITIVE  TO  NEGATIVE,  OR  THE  CONTRARY,  AND   THEN 

PROCEED  AS  IN  ADDITION.  (Art.  148.)  In  making  the  re- 
quired change,  it  will  be  expedient  to  alter,  in  some  instances, 
the  signs  of  the  numerator,  and  in  others,  the  sign  before  the 
dividing  line,  (Art.  143,)  so  as  to  leave  the  latter  always 
affirmative. 

Ex.   1.  From  i  subtract  _. 
b  ^" 

First  change  _,  the  fraction  to  be  subtracted,  to  I — 
m  m 

Secondly,  reduce  the  two  fractions  to  a  common  denomi- 
nator, making,  —  and  H 

bm  bm 

Thirdly,  the  sum  of  the  nmnerators  am  -  bh,  placed  over 

the  common  denominator,  gives  the  answer, Z 

bm 

2.  From  "+1  subtract  t    Ans.  ^^^id!!. 

r  d  dr 

3.  From  ^  subtract  ll^.     Ans.  ^Izl^^:^. 

my  my 

8 


58  ALGEBRA. 

4.  From  ^-±^,  subtract  ?£r!i      Ans.  ™:l2?. 

4  3  12 

5.  From  td  subtract  -*.      Ans.  ^iZ^^+^l 

m  y  my 

6.  From  ^5+1  subtract  izl.    7.  From:?  subtract  1. 

d  m  a  b 

152.  Fractions  may  also  be  subtracted,  like  integers,  by 
setting  them  down,  after  their  signs  are  changed,  without  re- 
ducing them  to  a  common  denominator. 

Fromi  subtract  -^^     Xns.'!L+'^. 
m  y  m      y 

In  the  same  manner,  an  integer  may  be  subtracted  from 
a  fraction,  or  a  fraction  fr'om  an  integer. 

From  a  subtract —     Ans.  a- — 
m  ra 

153.  Or  the  integer  ma}^  be  incorporated  with  the  fr-action, 
as  in  Art.  1 50. 

Ex.   1.  From  _  subtract  m.     Ans.  _  -  m=    ~^^W. 

y  y  y 

2.  From  4a+L  subtract  Sa-L    Ans.  ^i^^i^A^. 

c  d  cd 

«    -r.  iit-c      1,      ,c-b       .        d-l-'^b-2c 

3.  From  1+— — subtract Ans 1 

'  ■  [  d  d  d 

4.  From  a4-Sh  -  ^~^  subtract  Sa -  /i+^-. 

■'2  3 


MULTIPLICATION  OF  FRACTIONS. 

154.  By  the  definition  of  multiplication,  multiplying  b}^  a 
fraction  is  inking  a  part  of  the  multiphcand,  as  many  times  as 
there  are  like  parts  of  an  unit  in  the  multiplier.  (Art.  90.) 
Now  the  denominator  of  a  fraction  shows  into  what  parts  the 
integral  unit  is  supposed  to  be  divided  ;  and  the  numerator 
sliOAvs  how  many  of  those  parts  belong  to  the  given  fraction. 
In  multiplying  by  a  fraction,  therefore,  the  multiplicand  is 
to  be  divided  into  such  parts,  as  are  denoted  by  the  denom- 
inator ;  and  then  one  of  these  parts  is  to  be  ^repeated,  as 
oiany  times,  as  is  required  by  the  numerator. 


FRACTIONS.  59 


Suppose  a  is  to  be  multiplied  by  — 

A  fourth  part  of  a  is  — • 

4 

This  taken  3  times  is  -  i  ^_i  "L^^i.  (Art.  148.) 

4    "^4  '  4       4 

Again,  suppose  —  is  to  be  multiplied  by  — • 

One  fourth  of  ^is  ~.   (Art.  138.) 

(I       a      a      Sa 

This  taken  3  times  is  77+t7-1-t7=  t7> 

4b^4b  '  46     46 

the  product  required. 

In  a  similar  manner,  any  fractional  multiplicand  may  be 
divided  hito  parts,  by  multiplying  the  denominator  ;  and  one 
of  the  parts  may  be  repeated,  by  multiplying  the  numerator. 
We  have  then  the  following  rule  : 

155.  To  MULTIPLY  FRACTIONS,  MULTIPLY  THE  NUMERA- 
TORS TOGETHER,  FOR  A  NEW  NUMERATOR,  AND  THE  DENOMI- 
NATORS  TOGETHER,  FOR  A  NEW  DENOMINATOR. 

36  .        d  Sbd 

Ex.  1.  Multiply  —  into——      Product^-—. 

c  2m  zcm 

^   ,,  ,  .  1     a-\-d .  4/i        _.     ,         4ah-\-4dh 

2.  Multiply  — ^— into  -•     Product  4; — 

^  -^      y  m-z  my  -2y 

«    ,»,  n  .  1     (a-\-m)xh  .  4         _,     ,        (a-\-m)x4h 

3.  Multiply  ^  ^  J         into  .- r-    Product  V-/"^-^- 

^  -^  3  {a-n)  Sx{a-n) 

»,  n     a-\-h .        4-m       ^    ^^  .  "       1        .       3 

4.  Muh.  -4-, into— —     5.  Mult.  — -^  into^. 

S-\-d         c-\-y  a-\-Sr  8 

156.  The  method  of  multiplying  is  the  same,  when  there 
are  more  than  two  fractions  to  be  multiplied  together. 

a     c               771       _,     -         acm 
Multiply  together  p    t>    and Product   ^. 

For  ^X  ^  is,  by  the  last  article  ^,  and  this  into  !?  is  ^--. 
0      d  bd  y        bdy 

3.  Multiply ^,^,  'L,  and  _L.    Product  ^^  "-^H 
771      y      c  r  - 1  crnry  -  cmy 


60  ALGEBRA. 

3.  Mult.  5--y,  j  and  J-.    4.  Mult.  ^,  «Z_^,  and  |. 

157.  The  multiplication  may  sometimes  be  shortened,  by- 
rejecting  equal  factors,  from  the  numerators  and  denomina- 
tors. 

,    a  .       h        ^  c?     _.     -       dh 

1 .  Multiply  -  into  —  and  -.   Product  —  • 

^  -^  r  ay  ry 

Here  a,  being  in  one  of  the  numerators,  and  in  one  of  the 
denominators,  may  be  omitted.     If  it  be  retained,  the  product 

will  be  — .     But  this  reduced  to  lower  terms,  by  Art.  145, 
ary 

will  become  —  as  before. 
ry 

2.  Multiply  ^  into  !!!  and  «i.     Product  ^. 

"^^   m  ^a        U  6 

It  is  necessary  that  the  factors  rejected  from  the  numera- 
tors be  exactly  equal  to  those  which  are  rejected  from  the 
denominators.  In  the  last  example,  a  being  in  two  of  the 
numerators,  and  in  only  one  of  the  denominators,  must  be  re- 
tained in  one  of  the  numerators. 

3.  Multiply  f!±^  into  !>!?.     Product  '"''+'^"'. 

y  ah  ah 

Here,  though  the  same  letter  a  is  in  one  of  the  numerators, 
and  in  one  of  the  denominators,  yet  as  it  is  not  in  every  term 
of  the  numerator,  it  must  not  be  cancelled. 

4.  Multiply '!!?!+^ into i  and!!:. 

h  m         6a 

If  any  difficulty  is  found,  in  making  these  contractions,  it 
^vill  be  better  to  perform  the  multiphcation,  without  omitting 
any  of  the  factors  ;  and  to  reduce  the  product  to  lower  terms 
afterwards. 

158.  When  a  fraction  and  an  mfe^er  are  muUiplied  to- 
gether, the  numerator  of  the  fraction  is  multiphed  into  the 
integer.  Ti\e  denominator  is  not  altered ;  except  in  cases 
where  division  of  the  denominator  is  substituted  for  multipli- 
cation of  the  numerator,  according  to  Art.  139. 


FRACTIONS.  61 

^,  ,  wi     am      -p,,.  ^     a  ^  a  ^m     am 

Thus  aX— = —     lora=_;  and  _X— =  — 

y     y  1  1    y     2/ 

So  rX^X^=?^-^-^.     And.xl^l     Hence, 
d         o  Sd  bo 

159.  A  FRACTION  IS  MULTIPLIED  INTO  A  QUANTITY  EQUAL 
TO  ITS  DENOMINATOR,  BY  CANCELLING  THE  DENOMINATOR. 

Thus±Xb  =  a.     For  ^x6  =  — .      But  the  letter  h,  being 
h  b  h 

in  both  tlie  numerator  and  denominator,  may  be  set  aside. 

(Art.  145.) 

So  _^Lx  («-?/) -3m.     Andy:^X(3+m)=/i+3^. 
a-y  34-m 

On  the  same  principle,  a  fraction  is  muhiplied  into  any 
factor  in  its  denominator,  by  cancelling  that  fVictor. 

by  by      b  24  4 

160.  From  the  definition  of  multiplication  by  a  fraction,  it 
follows  that  what  is  commonly  called   a  compound  fraction,^ 

is  the  product  of   two  or  more  fractions.      Thus  '—  of  _  is 

4       0 

?X-.     For,  ?  of  ?,  is  1  of  ^  taken  three  times,  that  is, 
4     6  4       6       4       6 

^-uf  4-_^ .      But  this  is  the  same  as  -  multiplied  by  i. 
46^46^46  6  "^         *^  4 

(Art.  154.) 

Hence,  reducing  a  compound  fraction  into  a  simple  one,  is  the 
same  as  multiplying  fractions  into  each  other. 

Ex.  1.  Reduce   _  of  _iL.     Ans.       ^"^ 


7        6+2  76+14 

2.  Reduced  of  i  of  i+i.  Ans.    J^^ 

3        5       2a-m  S0a-15m 

3.  Reduce  i  of  i  of — I —  Ans.  ^ 


3        8-d  16S-2U 


*  By  a  compound  fraction  is  meant  a  fraction  of  a.  fraction,  and  not  a  fraction 
\vho6e  numerator  or  denominator  is  a  compound  quantity. 


62  ALGEBRA. 

161.  The  expression    fa,   lb,  %,   &c.  are  equivalent  to 
?^,    ^,   1^.     For  la  is  f  of  a,  which  is  equal  to  ^x«=^^. 

(Art.  158.)      Sol6=ix6^-. 


DIVISION  OF  FRACTIONS. 


162.  To  DIVIDE  ONE  FRACTION  BY  ANOTHER,  INVERT  THE 
DIVISOR,  AND  THEN  PROCEED  AS  IN  MULTIPLICATION.  (Alt. 
155.) 

Ex.1.  Divide   -by__.     Ans.       X-  =  r— 
b         d  b      c      be 

To  understand  the  reason  of  the  rule,  let  it  be  premised, 
that  the  product  of  any  fraction  into  the  same  fraction  inverted, 
is  always  a  unit. 

Thus    °x*-=!^=l.    And-^xi±^  =  l. 

b     a     ab  ti-\~y         a 

But  a  quantity  is  not  altered  by  multiplying  it  by  a  unit. 
Therefore,  if  a  dividend  be  multiplied,  first  into  the  divisor 
inverted,  and  then  into  the  divisor  itself,  the  last  product  will 
be  equal  to  the  dividend.  Now,  by  the  definition,  (art.  115,) 
"  division  is  finding  a  quotient,  which  multiplied  into  the  di- 
visor will  produce  the  dividend."  And  as  the  dividend  mul- 
tiplied into  the  divisor  inverted  is  such  a  quantity,  the  quo- 
tient is  truly  found  by  the  rule. 

This  explanation  will  probably  be  best  understood,  by  at- 
tending to  the  examples.  In  several  which  follow,  the  proof 
of  the  division  will  be  given,  by  multiplying  the  quotient  into 
the  divisor.  This  will  present,  at  one  viev/,  tlie  dividend 
multiplied  into  the  inverted  divisor,  and  into  the  divisor  itself. 

2.  Divide  !!L  by  ^±     Ans.  ,^  x|-=^. 

2d         y  2d     Sh     6dh 

Proof.   ''R  X-=—    the  dividend. 
Qdh     y       2d 

3.  Divide  ^  by  ^1.     Ans.  ^x  ^=^^ 

r  y  r        5d       5dr 

Proof.  ?H::!?yx-=^. 

5dr  y         r 


FRACTIONS.  63 

4.  Divide by Ans. X— 7-= — 

X  a  X       4hr     rx 

Proof.  ?^x  — =  --thecUvi(lend. 
♦  rx       a  X 

5.  Dmde  ^  by  i^.      Ans.   i^xl^=l^. 

o  10?/  o        IS/i       /i 

6.  Divide  ^^  by  ±lL.     7.  Divide  ^ZZIL^  by  .±-. 

163.  When  a  fraction  is  divided  by  an  integer,  the  denomi- 
nator of  the  fraction  is  muUipUed  into  the  mteger. 

Tiius  the  quotient  of  -  divided  by  m,  is  — 
^  b  bm 

For  m  =  —  :  and  by  the  last  article,  ^-i-_=_X— =1 — 
1  ^  .  b      I     b     m     bm 

So -i-/i= X-= And  _-^6=:— -=-. 

fl-6  a-b     h     ah-bh  4  2^     8 

In  fractions,  muUiplication  is  made  to  perfonn  the  office 
of  division  ;  because  division  in  the  usual  form  often  leaves  a 
troublesome  remainder  :  but  there  is  no  remainder  in  multi- 
plication. In  many  cases,  there  are  methods  of  shortening 
the  operation.  But  these  will  be  suggested  by  practice, 
without  the  aid  of  particular  rules. 

164.  By  the  definition,  (art.  49,)  "the  reciprocal  of  a 
quantity,  is  the  quotient  arising  from  dividing  a  unit  by  that 
quantity." 

Therefore  the  reciprocal  of  _  is  1-f-— =  1  X-~—   That  is, 

b  b  a     a 

The  reciprocal  of  a  fraction  is  the  fraction  inverted. 

Thus  the  reciprocal  of  — - —  is  "^"^^  ;    the  reciprocal  of 
m-\-y  b 

—  is  zR  or  Sy ;  the  reciprocal  of  k  is  4.  Hence  the  recip- 
rocal of  a  fraction  whose  numerator  is  1,  is  the  denominator 
of  the  fraction. 

Thus  the  reciprocal  of  _  is  a ;  of  -,  is  a4-b,  &c. 

a  a+o 


64  ALGEBRA. 

65.  A  fraction  sometimes  occurs  in  the  numerator  or  de- 
nominator of  another  fraction,  as  t^.    It  is  often  convenient, 

b 

in  the  course  of  a  calculation,  to  transfer  such  a  fraction, 
from  the  numerator  to  the  denominator  of  the  piincipal  frac- 
tion, or  the  contrary.  That  this  may  be  done  without  alter- 
ing the  value,  if  the  fraction  transferred  ])e  inverted^  is  evi- 
dent fromi  the  following  principles  : 

First,  Dividing  by  a  fraction,  is  the  same  as  multiplying  by 
the  fraction  inverted.     (Art.  162.) 

Secondly,  Dividing  the  numerator  of  a  fraction  has  the 
same  effect  on  the  value,  as  muliiplying  the  denominator ;  and 
multiplying  the  numerator  lias  the  same  effect,  as  dividing 
the  denominator.     (Art.  139.) 

Thus  in  the  expression  I-  the  numerator  of  _  is  multiplied 

X  X 

into  f.  But  the  value  will  be  the  same,  if,  instead  of  multi- 
plying the  numerator,  we  divide  the  denominator  by  |,  that  is, 
multiply  the  denominator  by  |. 

Therefore  i^=^.  So    'L=1^ 

X      ^x  gm     m 

AndJl^ ^ ^_i_      And^-^     ^^-^^ 


h-\-y     iX{h+y)      Ih-i-iy  hi  m 

166.  Multiplying  the  numerator,  is  in  effect  m.ultiplying  the 
value  of  the  fraction.  (Art.  137.)  On  this  principle,  a  frac- 
tion may  be  cleared  of  a  fractional  co-efficient  which  occurs 
in  its  numerator. 

Thus  L«=lx^=^.      And  i«=Lx^=  i 

h    6    h    bh  y    ^    y    ^y 

And  3^:1^ r^Lx^^^^.      And  £=^. 
m         3        m        3wi  5a     20a 

On  the  other  hand,  J!  =  _  x  —  =  — 

7a;     7      a;       a; 

»     ,    a      1  ,  «     xa        A     1       4a  4a 

And    —=  —  X-=—      And = -^ — 

Sy     S     y      y  5d-{~5x     d-{~x 

167.  But  multiplyhig  the  denominator,  hy  another  fraction, 
is  in  effect  dividing  the  value  ;  (Art.  138.)  that  is,  it  is  multi- 
plying the  value  by  the  fraction  inverted.  The  principal  frac- 
tion may  therefore  be  cleared  of  a  fractional  co-efficient, 
which  occurs  in  its  denominator. 


SliMPLE  EQUATIONS.  65 

Thus  -=^^5=^x5=-.     And±=Z?. 

ly  Sy  pn      4m 

On  the  other  hand,  —  =  - — 
Sx     ix 

And  h±^'=y±±.  And  ?^=^. 

167.  b.  The  numerator  or  the  denominator  of  a  fraction, 
may  be  itself  a  fraction.     Tiie  expression  may  be  reduced  to 
a  more  simple  form,  on  the  principles  which  have  been  applied 
in  the  preceding  cases. 
a 

b     a       c     ad 
^^'''c=b^d=Tc 


y      X  r      nr 

And  f— r~-  And  ~=~- 

h—hy  mm 


SECTION  VII. 


SIMPLE  EaUATIONS. 


Art.  168.  The  subjects  of  the  preceding  sections  are  m- 
troductory  to  what  may  be  considered  the  peculiar  province 
of  algebra,  the  investigation  of  the  values  of  unknown  quan- 
tities, by  means  of  equations. 

An  equation  is  a  proposition,  expressing  in  algebraic 

CHARACTERS,  THE  EQUALITY  BETWEEN  ONE  QUANTITY  OR  SET 
OF  QUANTITIES  AND  ANOTHER,  OR  BETWEEN  DIFFERENT  EX- 

9 


66  ALGEBRA. 

PRESSIONS  FOR  THE  SAME  QUANTITY.*       TllllS  X-\-a  =  h-\-C,i3 

an  equatioi),  in  which  the  snm  of  cc  and  a,  is  equal  to  the  sum 
of  b  and  c.  The  quantities  on  the  two  sides  of  the  sign  of 
equahty,  are  sometimes  called  the  members  of  the  equation  ; 
the  several  terms  on  the  left  constituting  the  first  member, 
and  those  on  the  nght,  the  second  member. 

169.  The  object  aimed  at,  in  what  is  called  the  resolution 
or  reduction  of  an  equation,  is  to  find  the  value  of  the  unknown 
quantity.  In  the  first  statement  of  the  conditions  of  a  problem, 
the  known  and  unknown  quantities  are  frequently  throAvn 
promise uousl}^  together.  To  find  the  value  of  that  which  is 
required,  it  is  necessary  to  bring  it  to  stand  by  itself,  while 
all  the  others  are  on  the  opposite  side  of  the  equation.  But 
in  doing  this,  care  must  be  taken  not  to  destroy  the  equation, 
by  rendering  the  two  members  unequal.  Many  changes 
may  be  made  in  the  arrangement  of  the  terms,  without  af- 
fecting the  equality  of  the  sides. 

170.  The  reduction  of  an  equation  consists,  jhen, 
in  bringing  the  unknown  quantity  by  itself,  on  one 
side,  and  all  the  known  quantities  on  the  other  side, 
withdut  destroying  the  equation. 

To  eflfect  this,  it  is  evident  that  one  of  the  members  must 
be  as  much  increased  or  diminished  as  the  other.  If  a  quan- 
tity be  added  to  one,  and  not  to  the  other,  the  equality  will 
be  destroyed.     But  the  members  will  remain  equal ; 

If  the  same  or  equal  quantities  be  added  to  each.     Ax.  1. 
If  the  same  or  equal  quantities  be  subtracted  fioni  each.  Ax.  2. 
If  each  be  multiplied  by  the  same  or  equal  quantities.   Ax.  3. 
If  each  be  divided  by  the  same  or  equal  quantities.     Ax.  4. 

171.  It  may  be  farther  observed  that,  in  general,  if  the 
unknown  quantity  is  connected  with  others  by  addition,  mul- 
tiplication, division,  &c.  the  reduction  is  made  by  a  contrary 
process.  If  a  known  quantity  is  added  to  tlie  unknown,  the 
equation  is  reduced  by  subtraction.  If  one  is  multiplied  by 
the  other,  the  reduction  is  eflfected  by  division,  &c.  The 
reason  of  this  will  be  seen,  by  attending  to  the  several  cases 
in  the  following  articles.  The  known  quantities  may  be  ex- 
pressed either  by  letters  or  figures.  The  unknovm  quantity 
is  represented  by  one  of  the  last  letters  of  the  alpliabet,  gen- 
erally x^  y,  or  z.     (Art.   27.)     The  principal  reductions  to 

*  See  Note  D, 


SIMPLE  EQUATIONS.  ^^ 

be  considered  in  this  section,  are  those  which  are  effected  by 
transposition,  multipUcation,  and  division.  These  ought  to  be 
made  perlectiy  faniihar,  as  one  or  more  of  them  will  be  ne- 
cessary, m  the  resolution  of  almost  every  equation. 

TRANSPOSITION. 
172.  In  the  equation 

the  number  7  being  connected  with  the  unknown  quantity  x 
by  the  sign  -,  the  one  is  subtracted  from  the  other.  To  re- 
duce the  equation  by  a  contrary  process,  let  7  be  added  to 
both  sides.     It  then  becomes 

a:-7+7  =  9+7. 

The  equality  of  the  members  is  preserved,  because  one  id 
as  much  increased  as  the  other.  (Axiom  1.)  But  on  one 
side,  we  have  -  7  and  +  7.  As  these  are  equal,  and  have 
contrary  signs,  they  balance  each  other,  and  may  be  cancel- 
led.     (Art.'^77.)     The  equation  wiU  then  be 

a;=9+7. 
Here  the  value  of  x  is  found.     It  is  sho^^^l  to  be  equal  to 
9-1-7,  that  is  to   16.     The   equation   is   therefore    reduced. 
The  unlvnown  quantity  is  on  one  side  by  itself,  and  all  the 
known  quantities  on  the  other  side. 

In  the  same  manner,  if  x-b  =  a 

Adding  b  to  both  sides  x  -  b-\-b  =  a-\-b 

And  cancelhng  (  -  b-\-b)  x=ia-{-b. 

Here  it  will  be  seen  that  the  last  equation  is  the  same  as 
the  first,  except  that  b  is  on  the  opposite  side,  with  a  contra- 
ry sign. 

Next  suppose  y^c  =  d. 

Here  c  is  added  to  the  unknown  quantity  y.     To  reduce  the 

equation  by  a  contrary  process,  let  c  be  subtracted  from  both 

sides,  that  is,  let  -  c,  be  applied  to  both  sides.     We  then  have 

y-\-c-Cz=zd-c. 

The  equality  of  the  members  is  not  affected,  because  one 
is  as  much  diminished  as  the  other.  When  (-^c-c)  is  can- 
celled, the  equation  is  reduced,  and  is  ^ 

y  =  d-c. 

This  is  the  same  as  y-\-c=d,  except  that  c  has  been  trans- 
posed, and  has  received  a  contrary  sign.  We  hence  obtain 
the  folloAving  general  nile  : 


68  ALGEBRA. 

173.  When  known  quantities  are  connected  with  the 

UNKNOWN  quantity  BY  THE  SIGN  -]-  OR  -  ,  THE  EQUATION  IS 
REDUCED  BY  TRANSPOSING  THE  KNOWN  QUANTITIES  TO 
THE  OTHER  SIDE,  AND  PREFIXTNl  THE  CONTRARY  SIGN. 

This  is  called  reducing  an  -  v.^.tion  by  addition  or  subtrac- 
tion, because  it  is,  in  etfect,  a.  .iing  or  subtracting  certain 
quantities,  to  or  from,  each  of  the  members. 
Ex.1.  Reduce  the  equation  x-\-^h  -  m=h  -  d 

Transposing-j-36,  we  have  x-m=h-d  -Sb 

And  transposing  -  m,  x=h-d-  Sb-\-m. 

174.  When  several  terms  on  the  same  side  of  an  equation 
are  alike,  they  may  be  united  in  one,  by  the  rules  for  reduc- 
tion in  addition.      (Art.  72  and  74.) 

Ex.  2.  Reduce  the  equation  x-\-5b  -  4h=7b 

Transposing  56  -  4/i  x=7b  -  5b-\-4:h 

Uniting  76  -  5b  in  one  term  a;:=26-|-4/i. 

175.  The  unknown  quantity  must  also  be  transposed, 
whenever  it  is  on  both  sides  of  the  equation.  It  is  not  mate- 
rial on  which  side  it  is  finally  placed.  For  if  rr=3,  it  is  evi- 
dent that  ^  =  x.  It  may  be  well,  however,  to  bring  it  on  that 
side,  where  it  will  have  the  affirmative  sign,  when  the  equa- 
tion is  reduced. 

Ex.  3.  Reduce  the  equation  <^x-\-2hz=zh-\-d-\-^x 

By  transposition  2h-h-d=Sx-  2x 

And  h-d=x. 

176.  When  the  same  term,  with  the  same  sign,  is  on  oppo- 
site  sides  of  the  equation,  instead  of  transposing,  we  may  ex- 
punge it  from  each.  For  this  is  only  subtracting  the  same 
quantity  from  equal  quantities.      (Ax.  2.) 

Ex.4.  Reduce  the  equation  x-]-Sh-\-d=b-\-Sh-\-7d 

Expunging  3A  x-\-d=b-\-ld 

And  x=b-\-6d. 

177.  As  all  the  terms  of  an  equation  may  be  transposed, 
or  supposed  to  be  transposed ;  and  it  is  immaterial  which 
member  is  written  first ;  it  is  evident  that  the  signs  of  all  the 
terms  may  be  changed,  without  atTecting  the  equality. 

Thu|^  if  we  have  x-b  =  d-  a 

Then  by  transposition  -  d-\-a=  -x  -\-b 

Or,  inverting  the  members  -  x-\-b=  -  d-\-a. 

1 78.  If  all  the  terms  on  one  side  of  an  equation  be  trans- 
posed, each  member  will  be  equal  to  0 


SIMPLE  EQUATIONS. 


G9 


Thus,  if  x-{-b  =  d,  then  cc-]-h  -  f/=0. 

It  is  frequently  convenient  to  reduce  an  equation  to  this 
form,  in  which  the  positive  and  negative  terms  balance  each 
other.     In  the  example  just  given,  x-\-b  is  balanced  by  -  d. 
For  in  tlil  first  of  the  two  equations,  x-\-b  is  ec^ual  to  d. 
Ex.  5.  Reduce  a-|-2.r- 8=6- 4+a,'-}-rt. 

6.  Reduce  y-\-ah  -  hm=aJ^2y  -  ab-\rhm. 

7.  Reduce  /i+30-|-7a;=8  -  6/i-i-6a;  -  d-\-b. 

8.  Reduce  bh-^2l -4x+d=l'2 -Sx^d-7bL 

REDUCTION  OF  EQUATIONS  BY  MULTIPLICATION. 

179.  The  unknovm  quantity,  instead  of  being  connected 
with  a  known  quantity  by  the'  sign  -|-  or  -,  may  be  divided 

by  it,  as  in  the  equation  — =  6. 
a 

Here  the  reduction  cannot  be  made,  as  in  the  preceding 
instances,  by  transposition.  But  if  both  members  be  multi- 
plied by  a,  (Art.  170,)  the  equation  will  become, 

x=ab. 

For  a  fraction  is  multiplied  into  its  denominator,  by  removing 
the  denominator.  This  has  been  proved  from  tlie  properties 
of  fractions.  (Art.  lo9.)  It  is  also  evident  from  the  sixth 
axiom. 

Thus  x=^-—- (jHi^lX^^^:^^,  &c.     For  in  each 
a  ■"  3  a-\-b  d^o 

of  these  instances,  x  is  both  multiplied  and  divided  by  the 
same  quantity;  and  this  makes  no  alteration  in  the  value. 

180.'  When  the  unknown  quantity  is  DIVIDED  by  a 

KNOWN  QUANTITY,    THE  EQUATION    IS    REDUCED  BY  MULTI- 
PLYING EACH  SIDE  BY  THIS  KNOWN  QUANTITY. 

The  same  transpositions  are  to  be  made  in  this  case,  as  in 
the  preceding  examples.  It  must  be  observed  also,  that  every 
term  of  the  equation  is  to  be  multiphed.  For  the  several 
terms  in  each  member  constitute  a  compound  multiplicand, 
which  is  to  be  multiplied  according  to  Art.  98. 
Ex.  1.  Reduce  the  equation  —]-a=b-{-d 

Multiplying  both  sides  by       c 

The  product  is  x-{-ac=bc-[-cd 

And  x=bc-{-cd-ac. 


70  ALGEBRA. 


2.  Reduce  the  equation  -j-5  =  20 

Multiplying  by  6  a;  -  4+30  =  1 20 

And  a:=  120+4 -^0=94. 

3.  Reduce  the  equation  -{-d=h 

a-\-b 

MuUiplying  by  a+6  (Art.  100.)     x-\-ad-\-bd=ah-]-bh. 
And  X  =  ah-\-  bh  -ad~bd. 

181.  When  the  unknown  quantity  is  in  the  denominator  of 
a  fraction,  the  reduction  is  made  in  a  similar  manner,  by  mul- 
tiplying the  equation  by  this  denominator. 

Ex.  4.  Reduce  the  equation  — +7=8 

10  -a; 

Multiplying  by  10  -  a;  6+70  -  7a:=80  -  8x 

And  a;=4. 

182.  Though  it  is  not  generally  necessary,  yet  it  is  often 
convenient,  to  remove  the  denominator  from  a  fraction  con- 
sisting of  knoion  quantities  only.  This  may  be  done,  in  the 
same  manner,  as  the  denominator  is  removed  from  a  fraction, 
which  contains  the  unknown  quantity. 

Take  for  exampl  e  - = -+  - 

a    b     c 

Multiplying  by  a  jr=  -  -| 

Multiplying  by  6  bx= ficZ+?-  i 

Multiplying  by  c  bcx=acd-\-abh. 

Or  w^e  may  multiply  by  the  product  of  all  the  denomina- 
tors at  once. 

In  the  same  equation  -—t-\ — 

a    b     c 

,,,.,.       ,        ,  abcx     abed  ,  abch 

Multiplying  by  abc 


a 


b     '      c 


Then  by  cancelhng  from  each  term,  the  letter  which  is 
common  to  its  numerator  and  denominator,  (Art.  145,)  we 
have  bcx=a€d-\-abh,  as  before.     Hence, 

183.  An  equation  may  be  cleared  of  FRACTIONS  by 

MULTIPLYlNCx  EACH  SIDE  INTO  ALL  THE  DENOMINATORS. 


SIMPLE  EQUATION.  71 


b 


e 


Thus  the  equation  ±  =_-|-: 

ci      d      g     m 

is  the  same  as  dg^nx=ahgin-\-adeni  -  adgh. 

And  the  eauation  ^=_4— +— 

is  the  same  as  80a:  =:  40+ 48+ 180. 

In  clearing  an  equation  of  fractions,  it  will  be  necessary 
to  observe,  that  the  sign  -  prefixed  to  any  fraction,  denotes 
that  the  whole  value  is  to  be  subtracted,  (Art.  142,)  which  is 
done  by  changing  the  signs  of  all  the  terms  in  the  numerator. 

1  he  equation  =c  — - 

X  r 

is  the  same  as  ar  -  dr=crx  -obx-\-2hmx-\-6nx. 

REDUCTION  OF  EQUATIONS  BY  DIVISION. 

184.  When  the  unknown  quantity  is  MULTIPLIED 

INTO    ANY   KNOWN  QUANTITY,  THE  EQUATION  IS  REDUCED  BY 
DIVIDING  BOTH  SIDES  BY  THIS  KNOWN  QUANTITY.  (Ax.  4.) 

Ex.  1.  Reduce  the  equation  ax-^b  -Sh=d 

By  transposition  ax=d-{-Sh  -  b 

^,       Dividing  by  a  ^^rf-[-3/t-6^ 

2.  Reduce  the  equation       2x=--  _+  46 

c      h 

Clearing  of  fractions      2chx=ah-cd-\-4bch 

Dividing  bv  2ch  ^^ah-cdJribch^ 

^    "  2ch 

185.  If  the  unknown  quantity  has  co-efficients  in  several 
terms,  the  equation  must  be  divided  by  all  these  co-efficients, 
connected  by  their  signs,  according  to  Art.  121. 

Ex.  3.  Reduce  the  equation  Sx-bx=a-d 

That  is,  (Art.  120.)  {S-b)xx=a-d 

Dividing  by  3 -  6  -     ^~^ 


3-6 

4.  Reduce  the  equation  ax-\-x=h-4: 

Dividing  by  a+1  x= 


a+1 


72  ALGEBRA. 

Ex.  5.  Reduce  the  equation      x-^—=±t^ 
^  h  4 

Clearing  of  fractions      4hx  -  4x=ah-{-dh -  46 

Dividing  by  4h-4         ^^cih-]-dh- 4b^ 

4h-4 

186.  If  an)^  quantity,  either  known  or  unknown,  is  found 
as  a  factor  in  every  tenn,  the  equation  may  be  divided  by  it. 
On  the  otlier  hand,  if  any  quantity  is  a  divisor  in  every  term, 
tile  equation  may  be  multiplied  by  it.  In  this  way,  the  factor 
or  divisor  will  be  removed,  so  as  to  render  the  expression  more 
simple. 

Ex.  6.  Reduce  the  equation  ax-\-^ab  =  ^ad-\-a 

Dividing  by  a  a:-}-36  =  6(Z+l 

And  a;=6d+l-36. 

7.  Reduce  the  equation  ?±1  -  -=!!liA. 

XXX 

Multiplying  by  x  (Art.  139.)  .r-f  1  -  h=zh  -  d 

And  x=h-d-\-b-'l. 

8.  Reduce  the  equation  a:X («4-^)  -  a-b  =  dx (<H~^) 

Dividing  by  a-\-b  (Art.  Il8.)a; -  l=d 
And  x=d-\-\. 


187.  Sometimes  the  conditions  of  a  problem  are  at  first 
stated,  not  in  an  equation,  but  by  means  of  a  proportion.  To 
show  how  this  may  be  reduced  to  an  equation,  it  will  be  ne- 
cessary to  anticipate  the  subject  of  a  future  section,  so  far  as 
to  admit  the  principle  that  "  when  four  quantities  are  in  geo- 
metrical proportion,  the  product  of  the  two  extremes  is  equal 
to  the  product  of  the  two  means  :"  a  principle  which  is  at 
the  foundation  of  the  Rule  of  Three  in  arithmetic.  See 
Arithmetic. 

Thus,  if  a  :  b  ::c  :  d,  then  ad— be. 

And  if  3  :  4  : :  6  :  8,  then  3x8=4x6.     Hence, 

188.  A  PROPORTION  IS  CONVERTED  INTO  AN  EQUATION  BY 
MAKING  THE  PRODUCT  OF  THE  EXTREMES,  ONE  SIDE  OF  THE 
equation;  AND  THE  PRODUCT  OF  THE  MEANS,  THE  OTHER  SIDE. 


SIMPLE  EQUATIONS.  73 

Ex.  1 .  Reduce  to  an  equation  ax  :  b  :  :  ch  :  d. 

The  product  of  the  extremes  is  adx 

The  product  of  the  means  is  bch 

The  equation  is,  therefore  adx  —  bch. 

2.  Reduce  to  an  equation  a-\-b  :  c::h-m:y, 

Tiie  equation  is  ay-\-by  =  ch-cm. 

189.  On  the  other  hand,  an  equation  may  be  con- 
verted INTO  A  PROPORTION,  BY  RESOLVING  ONE  SIDE  OF  THE 
EQUATION  INTO  TWO  FACTORS,  FOR  THE  MIDDLE  TERMS  OF 
THE  PROPORTION  :  AND  THE  OTHER  SIDE  INTO  TWO  FACTORS, 
»^0R  THE  EXTRE3IES. 

As  a  quantity  may  often  be  resolved  into  different  pairs  of 
factors  ;  (Art.  42,)  a  variety  of  proportions  may  frequently 
be  derived  from  the  same  equation. 

Ex.  1.  Reduce  to  a  proportion  abc  =  deh. 

The  side  abc  may  be  resolved  into    a  x  be,  or  ab  x  c^  or  ac  X  b. 
And  deh  may  be  resolved  into  dx^hy  or  de  X  h,  or  d/i  X  «?• 

Therefore  a:  d::eh  :  be  And  ac  :  dh::e  :  b 

And  ab  :  de::h  :  c  And  ac  :  d : :  eh  :  b,  &c. 

For  in  each  of  these  instances,  the  product  of  the  extremes 
is  abc,  and  the  product  of  the  means  deh. 

2.  Reduce  to  a  proportion  ax-\-bx=cd  -  ch 

The  first  member  may  be  resolved  into  a:X  («+^) 
And  the  second  into  cx{d-  h) 

Therefore  x  :  c:  :  d-h  :  a-\-b      And  d-h  :  x:  :  a-}-b  :  c,  &c. 

190.  If  for  any  term  or  terms  in  an  ecpiation,  any  other  ex- 
pression of  the  same  value  be  substituted,  it  i  -  manifest  that 
the  equality  of  the  sides  will  not  be  affected. 

Thus,  instead  of  1 6,  w^e  may  write  2  x8,  or  _,  or  25  -  9,  &c. 

4 

For  these  are  only  different  forms  of  expression  for  the  same 
quantity. 

191.  It  will  generally  be  well  to  have  the  several  steps,  in 
the  reduction  of  equations,  succeed  each  other  in  the  follow- 
ing order. 

First,  Clear  the  equation  of  fractions.   (Art.  183.) 
Secondl}^,  Transpose  and  unite  the  terms.  (Arts.  173,  4,  5.) 
Thirdly,  Divide  by  the  co-efiicients  of  the  unknown  quan- 
tity.  (Arts.  184,  5.) 

10 


74  ALGEBRA. 

EXAMPLES. 

1.  Reduce  the  equation  _  -j-6  =  --  -{-7 

Clearing  of  fractions  24a;+192  =  20a;+224 

Transp.  and  uniting  terms  4a;=32 
Dividing  by  4  x=S. 

2.  Reduce  the  equation  ^-\-h=----{-d 

a  be 

Clearing  of  fractions  bcx-\-abx  -  acx=abcd  -  abch 

D-   -v                                     abed  — abch 
ividnig  xz=z 

beJ^ab  -  ac 

3.  Reduce40-6a;-16  =  l£0-14a'.      Ans.   a:=12. 

4.  Reduce    ^:i^+i  =  20-^-Zj^.     Ans.  x=^^. 

2    ^3  2  4 

5.  Reduce    ^+^  =  20-1.     6.  Reduce  1^-4=5. 

3     5  4  X 

7.  Reduce    _1__2  =  8.  8.  Reduce   _^=1. 

x-\-4  a?4-4 

9.  Reduce    x4---^-=U.     10.  Reduce  ^+^ -^  =  I. 
^2^3  2^3      4     10 

11.  Reduce    iZL^+Go::    ^^"^^"^ 


o 
13.  Reduce    3:r+5±t^=5J-lI^^. 

13.  Reduce    ?£zi-3=l^I±+^. 

3  3      ^ 

14.  Reduce    21+?iLriI=?fri+2llL''£ 

^16  8  2 

15.  Reduce    3x-^-4=^dlli-i  . 

4  3  12 

16.  Reduce    Z£±^-i6±i£+6=?f±5. 

3  5^2 

17.  Reduce    Hzif  _!^=5-6x+!£±ll 

5  3  3 

18.  Reduce    x-^J^+4=^l^-^.lzl+ilZ^. 

5  2  7^5 


SIMPLE  EQUATIONS.  75 


19.  Reduce  ?^±I+I-Jll^=!^^ 
5x-\-4t  ,  18-  a;  ,  .  j  .  a 


20.  Reduce 


SOLUTION  OF  PROBLEMS. 

192.  In  the  solution  of  problems,  by  means  of  equations, 
two  things  are  necessary:  First,  to  translate  the  statement  of 
the  question  from  common  to  algebraic  language,  in  such  a 
manner  as  to  form  an  equation  :  Secondly,  to  reduce  this 
equation  to  a  state  in  which  the  unknown  quantity  will  stand 
by  itself,  and  its  value  be  given  in  known  terms,  on  the  op- 
posite side.  The  manner  in  which  the  latter  is  effected,  has 
already  been  considered.  The  former  will  probably  occasion 
more  perplexity  to  a  beginner  ;  because  the  conditions  of 
questions  are  so  various  in  their  nature,  that  the  proper  me- 
thod of  stating  them  cannot  be  easily  learned,  like  the  reduc- 
tion of  equations,  by  a  system  of  definite  rules.  Practice, 
however,  will  soon  remove  a  great  part  of  the  difficulty. 

193.  It  is  one  of  the  principal  peculiarities  of  an  algebraic 
solution,  that  the  quantity  sought  is  itself  introduced  into  the 
operation.  This  enables  us  to  make  a  statement  of  the  con 
ditions  in  the  same  form,  as  though  the  problem  were  ahead}'- 
solved.  Nothing  then  remains  to  be  done,  but  to  reduce  the 
equation,  and  to  find  the  aggregate  value  of  the  known  quan- 
tities. (Art.  53.)  As  these  are  equal  to  the  unknown  quantity 
on  the  other  side  of  the  equation,  the  value  of  that  also  is 
determined,  and  therefore  the  problem  is  solved. 

Problem  1.  A  man  being  asked  how  niuch  he  gave  for  his 
watch,  replied ;  If  you  multiply  the  price  by  4,  and  to  the 
product  add  70,  and  from  this  sum  subtract  50,  the  remain- 
der will  be  equal  to  220  dollars. 

To  solve  this,  we  must  first  translate  the  conditions  of  the 
problem,  into  such  algebraic  expressions  as  will  form  an  equa- 
tion. 

Let  the  price  of  the  watch  be  represented  by  x 

This  price  is  to  be  mult'd  by  4,  which  makes  Ax 

To  the  product,  70  is  to  be  added,  making  4a;-]- 70 

From  this,  50  is  to  be  subtracted,  making  4x-j-70-50 


76  ALGEBRA. 

Here  we  have  a  number  of  the  conditions,  expressed  in 
algebraic  terms  ;  but  have  as  j'et  no  equation.  We  must  ob- 
serve then,  that  by  the  last  condition  of  the  problem,  the  pre- 
ceding" terms  are  said  to  be  equal  to  220. 

We  have,  therefore,  this  equation     4a:-[-'70- 50=220 

Which  reduced  gives  x=50. 

Here  the  value  of  x  is  found  to  be  50  dollars,  which  is  the 
price  of  the  watch. 

194.  To  prove  whether  we  have  obtained  the  true  value  of 
the  letter  which  represents  the  unknown  quantity,  we  have 
only  to  substitute  this  value,  for  the  letter  itself,  in  the  equa- 
tion which  contains  the  first  statement  of  the  conditions  of 
the  problem ;  and  to  see  whether  the  sides  are  equal,  after 
the  substitution  is  made.  For  if  the  answer  thus  satisfies  the 
conditions  proposed,  it  is  the  quantity  sought.  Thus,  in  the 
preceding  example, 

The  original  equation  is  4a:-|-70  -  50=220 

Substituting  50  for  x,  it  becomes      4  X  50+70  -  50=220 
That  is,  -220=220. 

Prob.  2.  Wliat  number  is  that,  to  which,  if  its  half  be  add- 
ed, and  from  the  sum  20  be  subtracted,  the  remainder  will  be 
a  fourth  of  the  number  itself? 

In  stating  questions  of  this  kind,  where  fractions  are 
concerned,  it  should  be  recollected,  that  ix  is  the  same  as 

1;  that  !a:=?i',  &c.   (Art.  161.) 
3  5 

In  this  problem,  let  x  be  put  for  the  number  required. 
Then  by  the  conditions  proposed,        ^+o"  ~^^~T 
And  reducing  the  equation  x=16. 

Proof,  16+^-^-20=1^. 

^2  4 

Prob.  3.  A  father  divides  his  estate  among  his  three  sons, 
in  such  a  manner,  that. 

The  first  has  $1000  less  than  half  of  the  whole  ; 

The  second  has  800  less  than  one  third  of  the  whole ; 

The  third  has  600  less  than  one  fourth  of  the  whole  ; 

What  is  the  value  of  the  estate  ? 

If  the  whole  estate  be  represented  by  x^  then  the  several 

shares  will  be  ^  - 1000,  and  ^  -  800,  and  ^  -600. 

Z  o  4 


SIMPLE  EQUATIONS.  77 

And  as  these  constitute  the  whole  estate,  they  are  together 
equal  to  x. 

We  have  then  this  equation  -  -  1000+f  -  800+f  -  600=.t. 

Wliich  reduced  gives  rr= 28800. 

Proof  !5?22~1000+!55^-800+???^^-6OO=288OO. 

195.  To  avoid  an  unnecessary  introduction  of  unknown 
quantities  into  an  equation,  it  maybe  well  to  observe,  in  this 
place,  that  when  the  sum  or  difference  of  two  quantities  is 
given,  both  of  them  may  be  expressed  by  means  of  the  same 
letter.  For  if  one  of  the  tw^o  quantities  be  subtracted  from 
their  sum,  it  is  evident  the  remainder  will  be  equal  to  the 
other.  And  if  the  difference  of  two  quantities  be  subtracted 
from  the  greater,  the  remainder  will  be  the  less. 

Thus  if  the  sum  of  two  numbers  be  20 

And  if  one  of  them  be  represented  by  x 

The  other  will  be  equal  to  20  -  x. 

Prob.  4.  Divide  48  into  two  such  parts,  that  if  the  less  be 
divided  by  4,  and  the  greater  by  6,  the  sum  of  the  quotients 
will  be  9. 

Here,  if  x  be  put  for  the  smaller  part,  the  greater  wall  be 
48 -a:. 

By  the  conditions  of  the  problem  --\- — Zi:=9. 

4         6 

Therefore  a:=  12,  the  less. 

And  48-a:=36,  the  gi-eater. 

196.  Letters  may  be  employed  to  express  the  known  quan- 
tities in  an  equation,  as  well  as  the  unknow^i.  A  particular 
value  is  assigned  to  the  numbers,  when  they  are  introduced 
into  the  calculation  :  and  at  the  close,  the  numbers  are  re- 
stored.  (Art.  52.) 

Prob.  5.  If  to  a  certain  number,  720  be  added,  and  the 
sum  be  divided  by  125  ;  the  quotient  will  be  equal  to  7392 
di^ided  by  462.     What  is  that  number? 

Let  X—  the  numl:er  required. 

a=720  rf=7392 

6  =  125  /i=462 


78  ALGEBRA.  ♦ 

Then  by  the  conditions  of  the  problem  x-\-a_d 

b        h 

Therefore  ^^hd-ah 

h 

Restoringthe  numbers,  r.^(^^^X^Jgg)z(ZiQX462)^^ 

462 
197.  When  the  resolution  of  an  equation  brings  out  a 
negative  answer,  it  shows  that  the  value  of  the  unknow^n 
quantity  is  contrary  to  the  quantities  which,  in  the  statement 
of  the  question,  are  considered  positive.  See  Negative  Quan- 
tities.  (Art.  54,  &c.) 

Prob.  6.  A  merchant  gains  or  loses,  in  a  bargam,  a  certain 
sum.  In  a  second  bargain,  he  gains  350  dollars,  and,  in  a 
third,  loses  60.  In  the  end  he  finds  he  has  gained  200  dol- 
lars, by  the  three  together.  How  much  did  he  gain  or  lose 
bv  the  first  ] 

In  this  example,  as  the  profit  and  loss  are  opposite  in  their 
nature,  they  must  be  distinguished  by  contrary  signs.     (Art. 
57.)   If  the  profit  is  marked  -[-?  <^he  loss  must  be  -  . 
Let  x=  the  sum  required. 
Then  according  to  the  statement  a:-j-350--60=200 

And  a;=  -  90 

The  negative  sign  prefixed  to  the  answer,  shows  that  there 
w^as  a  loss  in  the  first  bargain  ;  and  therefore  that  the  proper 
sign  of  X  is  negative  also.  But  this  being  determined  by  the 
answer,  the  omission  of  it  in  the  com'se  of  the  calculation 
can  lead  to  no  mistake. 

Prob.  7.  A  ship  sails  4  d^rees  north,  then  13  S.  then  17 
N.  then  19   S.  and  has  finally  11  degrees  of  south  latitude. 
What  was  her  latitude  at  starting  1 
Let  x=  the  latitude  sought. 

Then  marking  the  northings  +?  ^^^  the  southings  - ; 

By  the  statement  a:+4-13-f-17-19=-ll 

And  x=0. 

The  answer  here  shows  that  the  place  from  which  the  ship 
started  was  on  the  equator,  where  the  latitude  is  nothing. 

Prob.  8.  If  a  certain  number  is  divided  by  12,  the  quo- 
tient, dividend,  and  divisor,  added  together,  will  amount  to 
64.     What  is  the  number  ? 


SIMPLE  EQUATIONS.  79 

Let  x=  the  number  soiisrht.  *■ 


'&' 


Then  _l_La;4-12=:64. 

12 

And  x-^^^  =  48. 

13 

Prob.  9.  An  estate  is  divided  among  fom*  children,  in  such 
a  manner  that 

The  first  has  200  dollars  more  than  |  of  the  whole, 
The  second  has  340  dollars  more  than  |  of  the  whole, 
The  third  has  300  dollars  more  than  ^  of  the  whole, 
The  fourth  has  400  dollars  more  than  I  of  the  whole. 
What  is  the  value  of  the  estate  1  Ans.  4800  dollars. 

Prob.  10.  What  is  that  number  which  is  as  much  less  than 
500,  as  a  fifth  part  of  it  is  greater  than  40 1         Ans.  450. 

Prob.  11.  There  are  two  numbers  whoye  difference  is  40, 
and  which  are  to  each  other  as  6  to  5.  What  are  the  nmn- 
bers  ]  Ans.  240  and  200. 

Prob.  12.  Three  persons,  .4,  B,  and  C,  draw  prizes  in  a 
lottery.  A  draws  200  dollars  ;  B  draws  as  much  as  ^,  to- 
gether with  a  third  of  what  C  draws ;  and  C  draws  as  much 
as  A  and  B  both.     What  is  the  amomit  of  the  three  prizes  1 

Ans.  1200  dollars. 

Prob.  13.  What  number  is  that,  which  is  to  12  increased 
by  three  times  the  number,  as  2  to  9  ?  Ans.  8. 

Prob.  14.  A  ship  and  a  boat  are  descending  a  river  at  the 
same  time.  The  ship  passes  a  certain  fort,  when  the  boat  is 
13  miles  below.  The  ship  descends  five  miles,  while  the 
boat  descends  three.  At  what  distance  below  the  fort  will 
they  be  together  ?  Ans.  32^  miles. 

Prob.  15.  IVhat  number  is  that,  a  sixth  part  of  which  ex- 
ceeds an  eighth  part  of  it  by  20  ]  Ans.  480. 

Prob.  16.  Divide  a  prize  of  2000  dollars  into  tAvo  such 
parts,  that  one  of  them  shall  be  to  the  other,  as  9  :  7. 

Ans.  The  parts  are  1125,  and  875. 

Prob.  17.  What  sum  of  money  is  that,  whose  third  part, 
fourth  part-,  and  fifth  part,  added  together,  amount  to  94  dol- 
lars? Ans.  120  dollars. 


80  ALGEBRA. 

Prob.  18.  Two  travellers,  A  and  B,  360  miles  apart,  travel 
towajds  each  other  till  they  meet.  Jl's  progress  is  10  miles 
an  horn*,  and  ^'s  8.  Ho^\rfar  does  each  travel  before  they 
meet  ?  Ans.  Jl  goes  200  miles,  and  B  160. 

Prob.  19.  A  man  spent  one  third  of  his  life  in  England, 
one  fourth  of  it  in  Scotland,  and  the  remainder  of  it,  which 
was  20  years,  in  the  United  States.  To  what  age  did  he 
hve  1  Ans.  to  the  age  of  48. 

Prob.  20.  What  nmiiber  is  that  \  of  which  is  greater  than 
\  of  it  by  96  1 

Prob.  21.  A  post  is  j  in  the  earth,  t  in  the  water  and  13 
feet  above  the  water.     What  is  the  length  of  the  post  1 

Ans.  35  feet. 

Prob.  22.  Wliat  number  is  that,  to  which  10  being  added, 
I  of  the  sum  will  be  Q^  1 

Prob.  23.  Of  the  trees  m  an  orchard,  f  are  apple  trees,  Yb 
pear  trees,  and  the  remainder  peach  trees,  which  are  20 
more  than  \  of  the  whole.  What  is  the  whole  number  in 
the  orchard  1  Ans.  800. 

Prob.  24.  A  gentleman  bought  several  gallons  of  Avine  for 
94  dollars;  and  after  using  7  gallons  himself,  sold  \  cf  the 
remainder  for  20  dollars.     How  many  gallons  had  he  at  first  1 

Ans.  47. 

Prob.  25.  A  and  B  have  the  same  income.  A  contracts 
an  annual  debt  amounting  to  ^  of  it ;  B  lives  upon  j  of  it ; 
at  the  end  of  ten  years,  B  lends  to  A  enough  to  pay  off  his 
debts,  and  has  1 60  dollars  to  spare.  What  is  the  income  of 
each  1  Ans.  280  dollars. 

Prob.  26.  A  gentleman  lived  single  \  of  his  whole  life  ; 
and  after  having  been  married  5  years  more  than  \  of  his 
Ufe,  he  had  a  son  who  died  4  years  before  him,  and  who 
reached  only  half  the  age  of  his  father.  To  what  age  did 
the  father  live  1  Ans.  84. 

Prob.  27.  What  number  is  that,  of  which  if  i,  ^,  and  f  be 
added  tos^ether  the  sum  will  be  73  ?  Ans.  84. 

Prob.  28.  A  person  after  spending  100  dollars  more  than  J 
of  his  income,  had  remaining  35  dollars  more  than  i  of  it. 
Required  his  income. 


SIMPLE  EQUATIONS.  81 

Piob.  29.   In  the  composition  of  a  quantity  of  gunpowder 
The  nitre  was  1 0  lbs.  more  than  §  of  the  whole, 
The  sulphur  41  lbs:.  less  than  ^  of  the  whole, 
The  charcoal  2  lbs.  less  th.an  \  of  the  nitre. 

What  was  the  amount  of  gunpowder  ]  Ans.  69  lbs. 
Prob.  30.  A  cask  which  held  146  gallons,  was  filled  with 
a  mixture  of  brandy,  wine,  and  water.  There  were  15  gal- 
lons of  wine  more  than  of  brandy,  and  as  much  water  as  the 
brandy  and  wine  together.  What  quantity  was  there  of 
each  ? 

Prob.  31.  Four  persons  purcliascd  a  farm  in  company  for 
4755  dollars  ;  of  which  B  paid  three  times  as  much  as  A  ; 
C  paid  as  much  as  A  and  B ;  and  D  paid  as  much  as  C  and 
B.     What  did  each  pay  ]         Ans.  317,  951,  1268,  2219. 

Prob.  33.  It  is  required  to  divide  the  number  99  into  five 
such  parts,  that  the  first  may  exceed  the  second  by  3,  be  less 
than  tiie  third  by  10,  gieater  than  the  fourth  by  9,  and  less 
than  the  fifth  by"' 16. 

Let  x=  the  first  part. 
Then  a: -  3=  the  second,  x-^=  the  fomth, 

a:4- 1 0  :=  the  third,  a:-f  1 6  =  the  fifth. 

Therefore  x-\-x  -  3+2'+ 1 0+a:  -  9-|-a:+ 16  =  99. 

And  x=\l. 
Prob.  33.  A  father  divided  a  small  sum  among  four  sons. 
The  third  had  9  shillings  more  than  the  fourth  ; 
The  second  had  12  shillings  more  than  the  third  ; 
The  first  had  18  shillings  more  than  the  second  ; 
And  the  whole  sum  was  6  shillings  more  than  7  times  the 
sum  which  the  youngest  received. 
What  was  the  sum  divided  1  An?.  1 53. 

Prob.  34.  A  farnier  had  two  flocks  of  slieep,  each  contain- 
ing the  same  number.  Having  sold  from  one  of  these  39, 
and  from  the  otlier  93,  he  finds  tv.  ice  as  many  remaining  in 
the  one  as  in  the  other.  How  many  did  each  flock  originally 
contain '{ 

Prob.  35.  An  express,  travelling  at  the  rate  of  60  miles  a 
day,  had  been  dispatched  5  days,  when  a  second  was  sent 
after  him,  travelling  75  miles  a  day.  In  what  time  will  the 
one  overtake  the  other  ]  Ans.  20  days. 

Prob.  36.  The  age  of  .^  is  double  that  of  B,  the  age  of  B 
triple  that  of  C,  and  the  sum  of  ail  their  age.g  140.  What  is 
the  age  of  each  ? 

11. 


82  ALGEBRA. 

Prob.  37.  Two  pieces  of  cloth,  of  the  same  price  by  the 
yard,  but  of  different  lengths,  were  bought,  the  one  for  five 
pounds,  the  other  for  6\.  If  10  be  added  to  the  length  of 
each,  the  sums  will  be  as  5  to  6.  Required  the  length  of  each 
piece. 

Prob.  38.  A  and  B  began  trade  with  equal  sums  of  money. 
The  first  year,  A  gained  forty  pounds,  and  B  lost  40.  The 
second  year,  c6f  lost  i  of  what  he  had  at  the  end  of  the  first, 
and  B  gained  40  pounds  less  than  twice  the  sum  which  A 
had  lost.  B  had  then  twice  as  much  money  as  A.  What 
sum  did  each  begin  with  ]  Ans.  320  pounds. 

Prob.  39.  What  number  is  that,  which  being  severally  ad- 
ded to  36  and  52,  vrill  make  the  former  sum  to  the  latter,  as 
3  to  4? 

Prob.  40.  A  gentleman  bought  a  chaise,  horse,  and  har- 
ness, for  360  dollars.  The  horse  cost  twice  as  much  as  the 
harness  ;  and  the  chaise  cost  twice  as  much  as  the  harness 
and  horse  together.     What  was  the  price  of  each  ? 

Prob.  41.  Out  of  a  cask  of  wine,  from  which  had  leaked 
i  part,  21  gallons  were  afterwards  drawn  ;  when  the  cask  was 
found  to  be  half  full.     How  much  did  it  hold  1 

Prob.  42.  A  man  has  6  sons,  each  of  whom  is  4  years  older 
than  his  next  younger  brother  ;  and  the  eldest  is  three  times 
as  old  as  the  youngest.     What  is  the  age  of  each  1 

Prob.  43.  Divide  the  number  49  into  two  such  parts,  that 
the  greater  increased  by  6,  shall  be  to  the  less  diminished  by 
11,  as  9  to  2. 

Prob.  44.  What  two  numbers  are  as  2  to  3  ;  to  each  of 
which,  if  4  be  added,  the  sums  v/ill  be  as  5  to  7  ? 

Prob.  45.  A  person  bought  two  casks  of  porter,  one  of 
which  held  just  3  times  as  much  as  the  other  ;  from  each  of 
these  he  drew  4  gallons,  and  then  found  that  there  were  4 
times  as  many  gallons  remahiing  in  the  larger,  as  in  the  other. 
How  many  gallons  were  there  in  each  1 

Prob.  46.  Divide  tlie  number  68  into  two  such  parts,  that 
the  difference  between  tlie  greater  and  84,  shall  be  equal  to 
3  tunes  the  difference  between  the  less  and  40. 

Prob.  47.  Four  places  are  situated  in  the  order  of  the  let- 
ters A.  B.  C.  D.     The  distance  from  Jl  to  Z)  is  34  miles. 


POWERS. 


3i 


The  distance  from  A  to  B  is  to  the  distance  from  C  to  />  as 
2  to  3.     And  i  of  the  distance  from  A  to  B,  added  to  half 
the  distance  from  C  to  Z),  is  three  times  the  distance  from 
B  to  C.     What  are  the  respective  distances] 
Ans.  From  A  to  B=12;  from  B  to  C=4;  from  C  to  D=IS. 

Prob.  48.  Divide  the  number  36  into  3  such  parts,  thafj 
of  the  first,  i  of  the  second,  and  J  of  the  third,  shall  be  equal 
to  each  other. 

Prob.  49.  A  merchant  supported  himself  3  years,  for  50 
pounds  a  year,  and  at  the  end  of  each  year,  added  to  that 
part  of  his  stock  which  was  not  thus  expended,  a  sum  equal 
to  one  third  of  this  part.  At  the  end  of  the  third  year,  his 
original  stock  was  doubled.     What  was  that  stock? 

Ans.  740  pounds. 

Prob.  50.  A  general  liaving  lost  a  battle,  found  that  he 
had  only  half  of  his  army-[-3600  men  left  fit  for  action  ;  i  of 
the  army-|-600  men  being  wounded  ;  and  the  rest,  who  w^ere 
s  of  the  whole,  either  slain,  taken  prisoners,  or  missing.  Of 
how  many  men  did  his  army  consist  1  Ans.  24000. 

For  the  solution  of  many  algebraic  problems,  an  acquaint- 
ance with  the  calculations  of  powers  and  radical  quantities  is 
required.  It  will  therefore  be  necessary  to  attend  to  these 
before  finishing  the  subject  of  equations. 


♦ 


SECTION  VIII. 

INVOLUTION  AND  POWERS. 


Art.  198.  WHEN  a  quantity  is  multiplied  into  IT 
SELF,  THE  PRODUCT  is  called  a  P01\:ER. 

Thus  2x2=4,  the  square  or  second  power  of  2 

2x2x2=8,  the  cube  or  third  power. 
2  X  2  X  2  X  2:^  1 6,  the  fourth  power,  &c. 

So  10x1 0=  1 00,  the  second  power  of  1 0. 

10x10x10=1000,  the  third  power. 
10x10x10x10=10000,  the  fourth  power,  &c 


84  ALGEBRA. 

And  axcfr=f^(ii  the  secojid  power  of  a 

aX(i'Xct=aaaf  the  third  power, 
a  X  «  X  «  X  «=«^«««j  the  fourth  power,  &c. 

199.  The  original  quantity  itself  though  not,  like  the  pow- 
ers proceeding  from  it,  produced  by  multiplication,  is  never- 
theless called  the  first  power.  It  is  also  called  the  root  of 
the  other  powers,  because  it  is  that  from  which  they  are  all 
derived. 

200.  As  it  is  inconvenient,  especially  in  the  case  of  high 
powers,  to  write  down  all  the  letters  or  factors  of  which  the 
powers  are  composed,  an  abridged  method  of  notation  is  ge- 
nerally adopted.  The  root  is  written  only  once  ;  and  then  a 
number  or  letter  is  placed  at  tiie  right  hand,  and  a  little  ele- 
vated, to  signify  how  many  times  the  root  is  employed  as  a 
^actor,  to  produce  the  power.  This  munber  or  letter  is  called 
the  index  or  exponent  of  the  power.  Thus  a^  is  put  for  ax« 
or  aa,  because  the  root  «,  is  twice  repeated  as  a  factor,  to 
produce  the  power  aa.  And  a^  stands  for  aaa ;  for  here  a 
is  repeated  three  times  as  a  factor. 

The  index  of  the  first  power  is  1  ;  but  this  is  commonly 
omitted.     Thus  a^  is  the  same  as  a. 

201.  Exponents  must  not  be  confounded  with  co-efficients. 
A  co-efficient  shows  how  often  a  quantity  is  taken  as  a  part 
of  a  whole.  An  exponent  shows  how  often  a  quantity  is 
taken  as  a  factor  in  a  product. 

Thus  Aa;=a-\-a-[-a-\-a.         But  a'^^ox^X^X^* 

202.  The  scheme  of  notation  by  exponents  has  the  pecu- 
liar advantage  of  enabling  us  to  express  an  unknown  power. 
For  this  purpose  the  index  is  a  letter,  instead  of  a  numerical 
figure.  In  the  solution  of  a  problem,  a  quantity  may  occur, 
which  we  knov/  to  be  some  power  of  another  quantity.  But 
it  may  not  be  yet  ascertained  whether  it  is  a  square,  a  cube, 
or  some  higher  power.  Thus  in  the  expression  a"",  the  index 
X  denotes  that  a  is  involved  to  sorae  power,  though  it  does  not 
determine  what  power.  So  If  and  d"  are  powers  of  h  and  d ; 
and  are  read  the  mth  power  of  6,  and  the  nth  power  of  d. 
When  the  value  of  the  index  is  found,  a  nmnher  is  generally 
substituted  for  the  letter.  Thus  if  ?7ir=3  then  b"'=zb^ ;  but 
if  m=5,  them  b"'  =  ¥. 

203.  The  method  of  expressing  powers  by  exponents  is 
also  of  great  advantage  in  the  case  of  compound  quantities. 


POAYERS.  85 

Thus  a-f//4-f/h  or  «,-|-6-p/'  or  (a^-Z^-f-i/)  %  is  (a+6-f  c^)  x 
(^a-{-b-\-d)x{a-\-b-i-d)  that  is,  the  cube  of  {a-\-b-\-d).     But 
this  involved  at  length  would  be 
a'-}-Sa'bJr^a'd-i-Sab'-\-6abd-{-3ad'Jrb''-\-^b'd-\-obd'-{-d\ 

204.  If  we  take  a  series*  of  powers  whose  indices  increase 
or  decrease  by  1,  we  shall  find  that  the  powers  themselves 
increase  by  a  common  multiplier^  or  decrease  by  a  common  di- 
visor;  and  that  this  multiplier  or  divisor  is  the  original  quan- 
tity from  which  the  powers  are  raised. 

Thus  in  the  series  aaaaa,     aaaa,     aaa,     aa,     a ; 

Or  a^  a"         a^      a^    a'  ; 

the  indices  counted  from  right  to  left  are  1,  2,  3,  4,  5;  and 
the  common  difference  between  them  is  a  unit.  If  we  be- 
gin on  the  right  and  multiply  by  a,  we  produce  the  several 
powers,  in  succession,  from  right  to  left. 

Thus  ax<J=«^  the  second  term.      And  a^x«=c^^ 

rt^  X«=f^^  tlie  third  term.  a"  x«=«%  &c 

If  we  begin  on  the  left,  and  divide  by  a. 

We  have  a^-f-a=a^  And  a^-f-a=a-. 

205.  But  this  division  may  be  carried  still  farther ;  and 
we  shall  then  obtain  a  new  set  of  quantities. 

Thus  a-f.az3^=l.    (Art. 128.)  L-^a=-L.     (Art.  163.) 
a  a  aa 

l-j-a=l  i-^a=J_,  &c. 

a  aa  aaa 

The  whole  series  then 

,111. 

IS  aaaaa,  aaaa,  aaa,  a  a,  a,  1,  _,  — ,  ,  olc. 

a    a a    aaa 

Or  a%  a\  a\  a",  a,  1,  _,    — ,    -,  &c. 
a     a"     cr 

Here  the  quantities  on  the  right  of  1,  are  the  reciprocals  of 
those  on  the  left.  (Art.  49.)  The  former,  therefore,  may  be 
properly  called  reciprocal  powers  of  a;  while  the  latter  may 
be  termed,  for  distinction's  sake,  direct  powers  of  a.  It  may 
be  added,  that  the  powers  on  the  left  are  also  the  reciprocals 
of  those  on  the  right. 

♦  Note. — The  term  series  is  applied(|o  a  number  of  quantities  succeeding: 
each  other,  in  some  regular  order.  It  is  not  confined  to  any  particular  law  ot 
increase  or  decrease. 


S6  ALGEBRA. 

For  1-^1=1  x-=a.   (Art.  162.)  And  l-f-L=a'. 
a  1  a^ 

1^i^  =  lX^^=a^  l^-\  =  a\&c. 

a^  1  a* 

206.  The  same  plan  of  notation  is  applicable  to  compound 
quantities.     Thus  from  a+^>  ^^'^  haye  the  series, 

(a+5)',(a+6)%  («+fc),  1,   ^i_^^,  ^.  ^.&c. 

207.  For  the  convenience  of  calculation,  another  form  of 
notation  is  given  to  reciprocal  powers. 

Accordmg  to  this,  _or_=a~^     And   — or  _=a~^. 
a       a}  aaa       a^ 

1  or  i  =a--\         J_  or  -  =  a-\  &c. 
aa      (f  aaaa       a* 

And  to  make  the  indices  a  complete  series,  with  1  for  the 

common  difference,  the  term  %r  1,  which  is  considered  as 

a 

no  power,  is  written  a". 

The  powers  both  direct  and  reciprocal*  then, 

Instead  of  aaaa,  aaa,  aa,  a,  —,  -,  —,  — , ,  &c. 

a    a  aa   aaa  aaaa 

Will  be  a^  a\  a\  a\  a\  a"\a-\  a-%  a-\  &c. 

Or  a+\  a-^\  a+\  a+^  a\  a-\  a-\  a-%  a"*,  &c. 

And  the  indices  taken  by  themselves  will  be, 

4-4,+3,+2,+  l,0,-l,-2,-3,-4,  &c. 

208.  The  root  of  a  power  may  be  expressed  by  more  let- 
ters than  one. 

Thus  aa  X ««,  or  aal^  is  the  second  power  of  aa. 
Andaax«aX««j  or  aaPisthe  third  power  of  aa,  &c. 

Hence  a  certain  power  of  one  quantity,  may  be  a  different 
power  of  another  quantity.  Thus  a*  is  the  second  power  of 
a^,  and  the  fomth  power  of  a. 

209.  All  the  powers  of  1  are  the  same.  For  Ix^j  or 
IXlXlj  &c.  is  still  1. 


See  Note  E. 


INVOLUTION.  87 


INVOLUTION. 

210.  Involution  is  finding  any  power  of  a  quantity,  by 
multiplying  it  into  itself.  The  reason  of  the  following  gene- 
ral rule  is  manifest,  from  the  nature  of  powers. 

MULTIPLF  THE  QUANTITY  INTO  ITSELF,  TILL  IT  IS  TAKEN 
AS  A  FACTOR,  AS  MANY  TIMES  AS  THERE  ARE  UNITS  IN  THE 
INDEX  OF  THE  POWER  TO  WHICH  THE  QUANTITY  IS  TO  BE 
RAISED. 

This  rule  comprehends  all  the  instances  which  can  occur 
in  involution.  But  it  will  be  proper  to  give  an  explanation 
of  the  manner  in  wliich  it  is  applied  to  particular  cases. 

211.  A  single  letter  is  involved,  by  giving  it  the  index  of 
the  proposed  power ;  or  by  repeating  it  as  many  times,  as  there 
are  units  m  that  index. 

The  4th  power  of  a,  is  a^  or  aaaa.     (Art.  198.) 

The  6th  power  of  y,  is  i/^  or  yyyyyy. 

The  nth  power  of  x,  is  x"  or  xxx.  ..n  tunes  repeated. 

212.  The  method  of  involving  a  quantity  which  consists 
of  several  factors,  depends  on  the  principle,  that  the  power  of 
the  product  of  several  factors  is  equal  to  the  product  of  their 
powers. 

Thus  {ayYz^a^  y\     For  by  Art.  210  ;  {ayy=ayXay. 
But  ayXtty  —  ayay=aayy=za^y''\ 
So  (bmxy=bmx  x  bmx  X  bmx=bbkmmmxxx=  ¥m^x^. 
And  {adyY  =  ady X ady x ady ...n  times = a^d^y"". 

In  finding  the  power  of  a  product,  therefore,  we  may  either 
involve  the  whole  at  once  ;  or  we  may  involve  each  of  the 
factors  separately,  and  then  multiply  their  several  powers  in- 
to each  other. 

Ex.  1.  The  4th  power  of  dhy,  is  {dhy)\  or  dVy\ 

2.  The  3d  power  of  46,  is  {4by,  or  4'b%  or  646^ 

3.  The  nth  power  of  6ad,  is  (Sad)",  or  6  «"</". 

4.  The  3d  poAver  of  37n x 2i/,  is  (3mx2l/)^  or  27ni^xSy\ 

213.  A  compound  quantity  consisting  of  temas  connected 
by -{-and-,  is  involved  by  an  actual  multiplication  of  its 
several  parts.     Thus, 


88  ALGEBRA. 

(a-\-by=za-{-by  the  first  power. 


a^-Lab 
-\-abJrb'' 

(a-\-by^=a^-{-2ab-\-b'^,  the  second  power  of  {a-\~b.) 


a^^2a'b-\-  ab 
+  a%-^2ab''+b 


2 

3 


(a-\-by^d'-\-Sa'bJr^ab^+^\  the  thhd  power. 
a+  b 


a'-i^Sa^b+Sd'b^-{~  ab^ 
+  a'bJr^d'b'-JrSab'-^b' 


fa-\-bY=a'-{-4db+6a!'b''-{-4ab^-\-b\  the  4th  power,  &c. 

2.  The  square  of  a-b,  is  a^ -2ab-^b'^. 

3.  The  cube  of  a-{-\,  is  d-\-Sa^-[-Sa-{-l. 

4.  The  square  of  a+6+/i,  is  a''-\-2ab-\-2ah+b^+2bh+h^ 

5.  Required  the  cube  of  a-{-2d-\-S, 

6.  Required  the  4th  power  of  b-\-2. 

7.  Required  the  5th  power  of  a:-f-l- 

8.  Required  the  6  th  poAver  of  1  -6.     . 

214.  The  squares  of  binomial  and  residual  quantities  occur 
so  frequently  in  dlgebraic  processes,  that  it  is  important  to 
make  them  famihar. 

If  we  muUiply  a~\-h  into  itself,  and  also  a-h. 

We  have  a-\-h  And  a-h 

a+h  CL-h 


a^-\-ah  (^  —  ah 

-fa/i+Zi'  ^ -ah+h"" 


a^_j_2a/i+/il  a'-2ah-JrK\ 

Here  it  will  be  seen  that,  in  each  case,  the  first  and  last 
terms  are  s(iuares  of  ft  and  h ;  and  that  the  middle  term  is 
twice  the  product  of  a  into  L     Hence  the  squares  of  bino- 


INVOLUTION.  89 

mial  and  residual  quantities,  without  multiplying  each  of  the 
terms  separately,  may  be  found,  by  the  following  proposition.* 

The  square  of  a  binomial,  the  terms  of  which  are 

BOTH  positive,  IS  EQUAL  TO  THE  SQUARE  OF  THE  FIRST  TERM 
4-TWICE  THE  PRODUCT  OF  THE  TWO  TERMS,  -j-THE  SQUARE 
OF  THE  LAST  TERM. 

And  the  square  of  a  residual  quantity,  is  equal  to  the 
square  of  the  lirst  term,*-  twice  the  product  of  the  two  terms, 
-{-  the  scjuare  of  the  last  term. 

Ex.   1.  The  square  of  2a-|-6,  is  4rt--|-4a6-|-6". 

2.  The  square  of  /i-f  1,  is  /r'-f  2/i+l. 

3.  The  square  of  ab-\-cd,  is  (rb'^-{-2abcd-\-c-d-. 

4.  The  square  of  6?/-f  3,  is  36</2-|-36j/-(-9. 

5.  The.  square  of  3d  -  h,  is  9^^'^  -  6c//i+/i^ 

6.  The  square  of  a  -  1,  is  a'^  -  2a-f-l 

For  the  method  of  finding  the  higher  powers  of  binomials, 
see  one  of  the  succeeding  sections. 

215.  For  many  purposes,  it  Avill  be  sufficient  to  express  the 
powers  of  compound  quantities  by  exponents,  without  an  actual 
multiplication. 


Thus  the  square  of  a+6,  is  a-^b\'\  or  (rt+6)^     Art.  203. 
The  nth  power  of  bc-\-8-\-x,  is  (6c4-8+.t)". 

In  cases  of  this  kind,  the  vinculum  must  be  drawn  over  all 
the  terms  of  which  the  compound  quantity  consists. 

216.  But  if  the  root  consists  of  several  factors,' the  vincu- 
lum which  is  used  in  expressing  the  power,  may  either  extend 
over  the  whole;  or  maybe  apphed  to  each' of  the  factors 
separately,  as  convenience  may  require. 


Thus  the  square  of  a--\-bXc-\-d,  is  either 

.2 


a-f6xc+c/|   ora-|-6|  xc-\-d\  . 

For,  the  first  of  these  expressions  is  the  square  of  the  pro- 
duct of  the  two  factors,  and  the  last  is  the  product  of  their 
squares.     But  one  of  these  is  equal  to  the  other.     (Art.  212.) 

The  cube  of  axb-\-d,  is  (axb-j-dy,  or  ctx(b-\-dy. 


Euclid's  Elements,  Book  II.  prop.  4. 
12 


90  ALGEBRA. 

217.  When  a  quantity  whose  power  has  been  expressed  by 
a  vinculum  a.id  an  index,  is  afterwards  involved  by  an  actual 
multiplication  of  the  terms,  it  is  said  to  be  expanded. 

Thus  (a-\-by,  when  expanded,  becomes  a^^2ab-\-b^. 
And  (a-\-b+hy,  becomes  d'Jf-2ab+2ah-\-b''-{-2bh-{-h\ 

18.  With  respect  to  the  sign  which  is  to  be  prefixed  to 
quantities  involved,  it  is  important  to\)bserve,  that  when  the 

ROOT  IS  POSITIVE,  ALL  ITS  POWERS  ARE  POSITIVE  ALSO  ;  BUT 
WHEN  THE  ROOT  IS  NEGATIVE,  THE  ODD  POWERS  ARE  NEGA- 
TIVE, WHILE  THE  EVEN  POWERS  ARE  POSITIVE. 

For  the  proof  of  this,  see  Art.  109. 

The  2d  power  of  -  a  is-|-a^ 
The  3d  power  is  -  a' 

The  4th  power  is        -\-  a* 
The  5th  power  is  -  a^,  &c. 

219.  Hence  any  odd  power  has  the  same  sign  as  its  root. 
Bat  an  even  power  is  positive,  whether  its  root  is  positive  or 
negative. 

Thus+ax+a=«'' 
And  -aX  -a=a^. 

220.  A  QUANTITY  WHICH  IS  ALREADY  A  POWER,  IS  INVOLV- 
ED BY  MULTIPLYING  ITS  INDEX,  INTO  THE  INDEX  OF  THE  POW- 
ER TO  WHICH  IT  IS  TO  BE  RAISED. 

1.  The  3d  power  of  a\  isa''^^=a\ 

For  a^=aa:  and  the  cube  of  aa  is  aax««Xtt«=<'fflf<^flOf«=a'; 
which  is  the  6  th  power  of  a,  but  the  3d  poAver  of  a\ 

For  the  further  illustration  of  this  rule,  see  Arts.  233,  4. 

2.  The  4th  power  of  «^6^  is  a^''*b^'''  =  a'n\ 

3.  The  3d  power  of  4  a^  is  64  a^x\ 

4.  The  4th  power  of  2a^x^x%  is  ISa'^X^l^^d*. 

5.  The  5th  power  of  (a+6)^  is  (a+6)  ' '. 

6.  The  nth  power  of  a^,  is  ft^". 

7.  The  Tith  power  of  {x  ~  yY,  is  {x-yY". 

8.  l^Il^r- ^a'-^2a^b^J^b\     (Art.  214.) 

9.  'Jx^y'  ^a'xb'.       10.   {a'b'h'y=a^b^h'*. 


INVOLUTION.  91 

221.  The  rule  is  equally  applicable  to  powers  whose  expo- 
nents are  negative. 
Ex.  1.  The  3d  power  of  a-%  is  ar^''^=a-\ 

For  fl-'^=i-,    (Art.  207.)     And  the   3d  power  of  this  is 
aa 

lxlxl  =  -^=-L=a-. 

aa     aa     aa      aaaaaa      a* 

2.  The  4th  power  of  a^'b-^  is  a'b-'\  or  ^^. 

3.  The  cube  of  2x"y-^,  is  Sx^"y'"". 

4.  The  square  of  6 ^ a;- S  is  b^x-". 

5.  The  nth  power  of  x-  "*,  is  x-""",  or 

222.  It  must  be  observed  here,  as  in  Art.  218,  tliat  if  the 
sign  which  is  prefixed  to  the  powder  be  -,  it  must  be  changed 
to  +j  whenever  the  index  becomes  an  even  number. 

Ex.  1.  The  square  of  -  a^,  is  +a^.  For  the  square  of 
-  a\  is  -  a^X  -  ^^  W'hich,  according  to  the  rules  for  the  signs 
in  multiplication,  is-f-^^- 

2.  Butthec«6eof-aMs-a^    For -a^X  -  a^X -a^= -fl'- 

3.  The  square  of  -  af,  is  +2?^". 

4.  The  nth  power  of  -  a\  is  -^a^". 

Here  the  power  will  be  positive  or  negative,  according  as 
the  number  wiiich  n  represents  is  even  or  odd. 

223.  A  FRACTION  is  involved  by  involving  both 

THE  NUMERATOR  AND  THE  DENOMINATOR. 

1.  The  square  of  -  is  —     For,  by  the  rule  for  the  multi- 

b      b^ 
plication  of  fractions,  (Art.  155.) 
a     a  _aa_a^ 
b     b~~bb~'¥ 

2.  The  2d,  3d,  and  nth  powers  of  -,  are  _»—    and  -L. 

a         a^  a^  a" 

3.  The  cube  of  —,  is  ?^". 

Sy  %lf 

4.  The  nth  power  of  — ,is 


ay"     ay" 


02  ALGEBRA. 

5.  The  square  of  Z^!><(^±I^^,  is  ^^:^±^. 

6.  The  cube  of  H^,  is  Z^~!      (Art.  221.) 

224.  Examples  of  binomials,  in  which  one  of  the  terms  is 
a  fraction. 

1.  Find  the  square  of  a;-[-^,  and  rr  -  J,  as  in  art.  214. 

1 

2 


a?+ J  X  - 

rr+-|  x-^ 


x'+x+l  x'-x-\-l 

2.  The  square  of  a  +?,  is  a"-]-—^-. 

3.  The  square  of  a-(— ,  is  a;'^-(_6a:4--. 

4.  The  square  of  a;  -  A,  is  r^  -  ?^+  .^' 


— »»9  ®  0«w—  ■ 

225.  It  has  been  shown,  (Art.  165,)  t\mt  a.  fractional  co- 
efficient may  be  transferred  from  the  nvunerator  to  the  de- 
nominator of  a  fraction,  or  from  the  denominator  to  the  nu- 
merator. By  recurring  to  the  scheme  of  notation  for  recip- 
rocal powers,  (Art.  207,)  it  will  be  seen  that  any  factor  may 
also  be  transferred,  if  the  sign  of  its  index  be  changed. 

1.  Thus,  in  the  fraction  — ,  we  may  transfer  x  from  the 

y 

numerator  to  the  denominator. 
For^=^X^=^xl=~. 

y      y  y    ^'    y^' 

2.  In  the  fraction  — ,  we  may  transfer  y  from  the  deno- 

by' 
minator  to  the  numerator. 


by'    b     f     b    '  b 


INVOLUTION.  93 

da~*         d  b        by~'' 

a;-*        x^a*  ay  a 

226.  In  the  same  manner,  we  ma)^  transfer  a  factor  which 
has  a  positive  index  in  the  numerator,  or  a  negative  index  in 
the  denominator. 

ax^         a  .  .  _ 

1.    Thus  —r-  =  r-zr'    For  x^  is  the  reciprocal  of  .t  ', 

1  CiX  Ot 

(Arts.  205,  207,)  that  is,  x^=^—=t'     Therefore,  — r-^r-rj- 

'^'  by-^-   b  •  '^-  xy-'-  xd-'- 

227.  Hence  the  denominator  of  any  fraction  may  be  en- 
tirely removed,  or  the  numerator  may  be  reduced  to  a  unit, 
without  ahering  the  value  of  the  expression. 

a       1 
1.  Thus  T^r-TTi,  or  ah~\ 
b     ba    ' 

x-^         1 

x'a-""  1 


ADDITION  AND  SUBTRACTION  OF  POWERS. 

228.  It  is  obvious  that  powers  may  be  added,  like  other 
quantities,  by  writing  them  one  after  another  with  their  signs. 
(Art.  69.) 

Thus  the  sum  of  a^  and  6%  is  a^-j-6*. 

And  the  sum  of  a"  -  ¥  and  h^  -d\  is  a"  -6"+^'  -  d\ 

229.  The  same  powers  of  the  same  letters  are  like  quantities; 
(Art.  45,)  and  their  co-efficients  maybe  added  or  subtracted, 
as  in  Arts.  72  and  74. 

Thus  the  sum  of  2a^  and  3a^  is  oa^. 

It  is  as  evident  that  twice  the  square  of  a,  and  three  times 
the  square  of  a,  are  five  times  "the  square  of  a,  as  that  tAvice 
a  and  three  times  a,  are  five  times  a. 


94  ALGEBRA. 

To  -3aY  Sb""           Say  -6a^h^           ^(a+t/)' 

Add  -2rry  66"'        -7ay  Ga^/i^           4(a+i/)^ 

Sum  -5xhf                        -4ay  7(a+7/)'' 


230.  But  powers  of  different  letters  and  different  powers  of 
the  sawie  leltery  must  be  added  by  writing  them  down  with 
their  signs. 

The  sum  of  a^  and  a^  is  a^-{-a^. 

It  is  evident  that  the  square  of  «,  and  the  cube  of  a,  are 
neither  twice  the  square  of  a,  nor  twice  the  cube  of  a. 
The  sum  of  a^b""  and  Sa'b\  is  a'b"-{-Sa'b\ 

231.  Subtraction  of  powers  is  to  be  performed  in  the  same 
manner  as  addition,  except  that  the  signs  of  the  subtrahend 
are  to  be  changed  according  to  Art.  82. 

From  2a'  -Sb"  SJrb'  a'b"  5(a-hy 

Sub.  -6a*  46"  4/i^6«  a^b"  ^{a-hy 

Diff.  Sa'  -m'  S{a-hy 


MULTIPLICATION  OF  POWERS. 

232.  Powers  may  be  muUiphed,  hke  other  quantities,  by 
writing  the  factors  one  after  another,  either  witii,  or  without, 
the  sign  of  muUiphcation  between  them.     (Art.  93.) 

Thus  the  product  of  a^  into  ¥,  is  a^6^,  or  aaabb. 
MuU.       x-^  h'^b-"  Say  dh^x-"         a^by 

Into  a*"  ft*  -2x  4by'  a^b'^y 


Prod.       a"a;-2  -da^ry''  d'bya^b''y 


The  product  in  the  last  example,  may  be  abridged,  by 
bringing  together  the  letters  which  are  repeated. 

It  will  then  become  a^y 

The  reason  of  this  will  be  e\adent,  by  recmring  to  the  se- 
ries of  powers  in  Art.  207,  viz. 

o+S  a+\  a+\  a+\  a\  a-\  a-^  a-\  a-\  &c. 

Or,  which  is  the  same, 

1111 

actaa,    aaa,    aa,    a,    1,  — ,    — ,    — ,    ,  &c. 

'  *        '      '      ^    a^     ad*    aaa'    aaaa^ 


POWERS.  95 

By  comparing  the  several  terms  with  each  other,  it  will 
be  seen  that  if  any  two  or  more  of  them  be  multiplied  to- 
gether, their  product  will  be  a  power  whose  exponent  is  the 
sum  of  the  exponents  of  the  factors. 

Thus  a^  Xct'^=ciax  acia — aaaaa = a^ 

Here  5,  the  exponent  of  the  product,  is  equal  to  2-|-3,  the 
sum  of  the  exponents  of  the  factors. 

So  a"  Xa'"=«"^'". 

For  a",  is  a  taken  for  a  factor  as  many  times  as  there  are 
units  m  n ; 

And  a"*,  is  a  taken  for  a  factor  as  many  times  as  diere  are 
units  in  m  ; 

Therefore  the  product  must  be  a  taken  for  a  factor  as 
•Tiany  times  as  there  are  units  in  both  m  and  n.     Hence, 

233.  Powers  of  the  same  root  may  be  multiplied, 
by  adding  their  exponents. 

Thus  a''xa'=cr+'=a\     And  x'xx''Xx=x^'f=3f. 
Mult.     4a''       Sx'       by  arby  (^+^-2/)" 

Into       2a"       2x^       b'y  aWy  b-{-h-y 


Prod.    8a:"  by  (6+A-i/)'H-» 


Mult.  x^-\-a^y-{-Tif-\-y^  inio  x-y.     Ans.x^-y*. 
Mult.  4x'^y-\-3xy  -  1  into  2ar^  -  x. 
Midt.  a^-\-x-  5  into  2x^+0;+ 1. 

234.  The  rule  is  equally  applicable  to  powers  whose  expo- 
nents are  negative. 

1.  Thusa-'X«~'=«"'.      That  is  Ix- ^  ^ 


aa     aaa     aaaaa 

2.  y-"xy~'"=y~"~'"-      That  is  JLx— =-!-. 

2/2/"*    y"y'" 

3.  -a-'xa~'=-rt"'.     4.  a-^Xa'=a'-^=aK 
5.  a-"xa'"=<*"'""-  6.  ?/-'x2/^=2/'— 1- 

235.  If  a-\-b  be  multiplied  into  a  -  b,  the  product  will  be 
0^-6'  :  (Art.  110,)  that  is 


06  ALGEBRA. 

The   PRODjrT   of   the   sum  and  difference  of  two 

QUANTITIES,      IS     EQUAL     TO     THE     DIFFERENCE     OF     THEIR 
SQUARES. 

This  is  another  instance  of  the  facihty  with  which  general 
truths  are  demonstrated  in  algebra.     See  Arts.  23  and  77. 

If  the  sum  and  difference  of  the  squares  be  multiphed, 
the  product  will  be  equal  to  the  difference  of  the  fourth 
powers,  &c. 

Thus  (a,-y)x{a-\-y)==a:'-y\ 

{a'-y^)X{a'Jry')=a'-y\  &c. 


DIVISION  OF  POWERS. 

236.  Powers  may  be  divided,  hke  other  quantities,  by  re- 
jecting from  the  dividend  a  factor  equal  to  the  divisor  ;  or  by 
placing  the  divisor  under  the  dividend,  in  the  form  of  a  frac- 
tion. 

Thus  the  quotient  of  a^b^  divided  by  b\  is  a^.   (Art.  116.) 

Divide       9ay         126V         d'h+Say         dx{a^h+yy 
By  -3a^  2b'  a^  (^a-h-\-yy 


Quot.       -Sy'  b-i-Sif  d 


.    a^ 
The  quotient  of  a^  divided  by  a^,  is  _ .      But  this  is  equal 

to  a^     For,  in  the  series 

a+*,  a^\  a'^%  a'^\  a",  a~',  a~^,  a~%  a~\  &c. 

if  any  term  be  divided  by  another,  the  index  of  the  quotient 
will  be  equal  to  the  difference  betw^een  the  index  of  the  divi- 
dend and  that  of  the  divisor. 

Thus  a'^a'="'^^^^=a\'   And  a--f-a''=:^=a«-''. 

aaa  a" 

Hence, 

237.  A  POWER  MAY  BE  DIVIDED  BY  ANOTHER  POWER  OF 
THE  SAME  ROOT,  BY  SUBTRACTING  THE  INDEX  OF  THE  DI- 
VISOR   FROM    THAT    OF    THE    DIVIDEND. 


POWERS.  97 


Thus  y^-i.y''=y^=y\      That  is  ^^^y. 

yy 

Anda''+'-^a=a''+'-'  =  a\    That  is  ^=:a\ 


And  x"-i-x"=x"-^  =  x'z=\.    That  is  1=1. 

x" 

Divide  !/""•         6'         8a''+"         a"-^'         12(6+i/)' 
By         2/"  b'         ia'"  a'  Hb-^yY 


Quot,     t/'^  2a"  4(^*4-1/)"-' 

238.  Tiie  rule  is  equally  applicable  to  powers  whose  ex- 
ponents are  negative. 

The  quotient  of  a~^  by  a~^,  is  a~^\ 

Thit  is  _L_~_L=_L_x— --^^---. 

aaaaa     aaa     aaaaa       1       aaaaa     aa 

2.  -  x-^'-^a;-^:.^  -  a:-l     That  is  _L  -=-!_=  _£!_ = J_. 

-a;'  '  x'      -x'     -a^ 

3.  h'^h-'  =  h^+'  =  h\     That  is  h'^l=.h'x-=h'^ 

h  1 

4.  6a''-^2a-^=3a''+^  5.  6a^-:-a=6al 
6.  h'^¥=h'-'=h-\  7.  a^~rt'  =  «-l 

9.   {hJrxY-7-{h+x)  =  {h-\-xy-\ 

The  multiplication  and  division  of  powers,  by  adding  and 
subtracting  their  indices,  should  be  made  very  familiar  ;  as 
they  have  numerous  and  important  applications,  in  the  high- 
er branches  of  algebra, 

EXAMPLES  OF  FRACTIONS  CONTAINING  POWERS. 

239.  In  the  section  on  fractions,  the  following  examples 
were  omitted  for  the  sake  of  avoidijig  an  anticipation  of  the 
subject  of  powers. 

1.  Reduce  ^  to  lower  terms.     Ans.  ^^^ . 

3a-  3 

For  ^^oaMaa^6cta       .^^^    ^^^  . 
3a'       3aa         3         ^  '' 

2.  Reduce  — ^  to  lower  terms.      Ans.  ^  or  2x. 

3x'  1 

13 


98  ALGEBRA. 

3.  Reduce  ^!!l±^  to  lower  terms.     Ans.  ?±±i^. 

4.  Reduce  — ^_I__1_Air — M.  to  lower  terms. 

6a'y-\.4af 

Ans.     ^  ~    ^y+  y_  obtained  by  dividing  each  term  by  2ay. 

5.  Reduce—  and — ^,  to  a  common  denominator. 

a^         a~ 

a*  X^""*  is  a~^  the  first  numerator.      (Art.  146.) 
a^  Xfl~^  is  tt°=l,  the  second  numerator, 
a'  X<i~*  is  rt""^  the  common  denominator. 

The  fractions  reduced  are  therefore and 

6.  Reduce  and  _,  to  a  common  denormnator 

Ans.  I^:  and  '^,  or  %  and  _^.     (Art.  145.) 

da^         da'       5a^  5ar 

7.  Multiply  _,mto_,    Ans._.=_. 

8.  Multiply  ^^,  into"^^ 

9.  Multiply  'l±l_,into^-lJ. 

x-^  x-{-a 

10.  Multiply  il  into  —1,  and-?l. 

11.  Divided,  by ^.     Ans.^'=i*. 

,/         y'  ay      y 

12.  Divide  ^''"''^',  by  ^Iz^l'. 


13.  Divide  ^-y"\  by  ^!i±- 

2/  y 

14.  Divide  ^\  by  ?+]  . 

a  a 


RADICAL  QUANTITIES.  99 


SECTION  IX. 


EVOLUTION  AND  RADICAL  QUANTITIES.^ 


Art.  240.  If  a  quantity  is  multiplied  into  itself,  the  pro- 
duct is  a  power.  On  the  contrary,  if  a  quantity  is  resolved 
into  any  number  of  equal  fLictors,  each  of  these  is  a  root  of 
that  quantity. 

Thus  b  is  the  root  of  bbb;  because  bbb  may  be  resolved 
into  the  three  equal  factors,  b,  and  6,  and  b. 

In  subtraction,  a  quantity  is  resolved  into  two  parts. 

In  division,  a  quantity  is  resolved  into  two  factors. 

In  evolution,  a  quantity  is  resolved  into  equal  factors. 

241.  A  ROOT  OF  A  QUANTITY,  THEN,  IS  A  FACTOR,  WHICH 
MULTIPLIED  INTO  ITSELF  A  CERTAIN  NUMBER  OF  TIMES,  WILL 
PRODUCE  THAT  QUANTITY. 

The  number  of  times  the  root  must  be  taken  as  a  factor, 
to  produce  the  given  quantity,  is  denoted  by  the  name  of  the 
root. 

Thus  2  is  the  4th  root  of  16  ;  because  2x2x2x2  =  16, 
where  two  is  taken  four  times  as  a  factor,  to  produce  16. 

So  a^  is  the  square  root  of  a^ ;  for  a^Xa^=a\    (Art.  233.) 

And  a^  is  the  cube  root  of  a^ ;  for  frx^'X^^^a". 

And  a  is  the  6th  root  of  a^;  for  aX^X^XaXaX^^a". 

Powers  and  roots  are  correlative  terms.  If  one  quantity 
is  a  power  of  another,  the  latter  is  a  root  of  the  former.  As 
b^  is  the  cube  of  b,  b  is  the  cube  root  of  b^. 

242.  There  are  two  methods  in  use,  for  expressing  the 
roots  of  quantities  ;  one  by  means  of  the  radical  si^n  \/>  and 
the  other  by  a  fractional  index.  The  latter  is  generally  to 
be  preferred ;  but  the  former  has  its  uses  on  particular  occa- 
sions. 


♦  Newton's   Arithmetic,   Maclaurin,   Emerson,  Euler,   Saunderson.  and 
Simpson.  ' 


100  ALGEBRA. 

Wlien  a  root  is  expressed  by  tlie  radical  sign,  the  sigii  is 
placed  over  the  given  quantity,  in  this  manner,  \/a. 
Thus  Xya  is  the  2d  or  square  root  of  a. 
\/a  is  the  3d  or  cube  root. 
\^a  is  the  nth  root. 
And  \/a-{-y  is  the  nth  root  of  a-\-y. 

243.  The  figure  placed  over  the  radical  sign,  denotes  the 
number  of  factors  into  which  the  given  quantity  is  resolved  ; 
in  other  words,  the  number  of  times  the  root  nuist  be  taken 
as  a  factor  to  produce  the  given  quantity. 

So  that  \/ax\/o>=0" 

And       V«XV^XV^=«- 

And       \/ax\/(f"--'''^^^^^^^^  =(^- 

The  figure  for  the  square  root  is  commonl}'^  omitted ;  \^a 
being  put  for  \/a.  Whenever,  therefore,  the  radical  sign  is 
used  without  a  figure,  the  square  root  is  to  be  understood. 

244.  When  a  figure  or  letter  is  prefixed  to  the  radical  sign, 
without  any  character  between  them,  the  two  quantities  are 
to  be  considered  as  multiplied  together. 

Thus  2/\/a,  is  2xV^>  ^^^^^  i^j  ^  multiplied  into  the  root  of 
o,  or,  w^hich  is  the  same  thing,  twice  the  root  of  a. 

And  a:\/6,  is  xx  V^>  or  x  times  the  root  of  h. 

When  no  co-efficient  is  prefixed  to  the  radical  sign,  I  is 
always  to  be  understood ;  \fa  being  the  same  as  l\/rt,  that 
is,  once  the  root  of  a. 

245.  The  method  of  expressing  roots  by  radical  signs,  has 
no  very  apparent  connection  with  the  other  parts  of  the 
scheme  of  algebraic  notation.  But  the  plan  of  indicating 
them  hy  fractional  indices,  is  derived  directly  from  the  mode 
of  expressing  powers  by  integral  indices.  To  explain  this, 
let  a°  be  a  given  quantity.  If  the  index  be  divided  into  any 
number  of  equal  parts,  each  of  these  will  be  the  index  of  a 
root  of  a^. 

Thus  the  square  root  of  a^  is  a^.  For,  according  to  the 
definition,  (Art.  241,)  the  square  root  of  a^  is  a  factor,  which 
multiplied  into  itself  will  produce  a^.  But  a^X«^=«^-  (Art. 
233.)  Therefore,  a^  is  the  square  root  of  a^.  The  index  of 
the  given  quantity  a^  is  here  divided  into  the  two  equal 
parts,  3  and  3.  Of  course,  the  quantity  itself  is  resolved  into 
the  two  equal  factors,  a^  and  a^. 


RADICAL  QUANTITIES.  101 

The  cube  root  of  a^  is  d\     For  a^X0''^Xa^=0'^' 

Here  the  index  is  divided  into  three  equal  parts,  and  the 
quantity  itself  resolved  into  three  equal  factors. 

The  square  root  of  a^  is  a^  or  a.     For  ax«=«^- 

By  extending  the  same  plan  of  notation,  fractional  indices 
are  obtained. 

Thus,  in  taking  the  square  root  of  a^  or  a,  the  index  1  is 

divided  into  two  equal  parts,  J  and  k  ;  and  the  root  is  a^' 

On  the  same  principle, 

The  cube  root  of  a,  is  a^  =^\/a. 

± 
The  nth  root,  is  a''=\/a,  &c. 

And  the  nth  root  of  a-j-a-,  is  (a-{-x)"=\/a-\-x. 

246.  In  all  these  cases,  the  denominator  of  the  fractional 

index,  expre&ses  the  number  of  factors  into  which  the  given 

quantity  is  resolved. 

X        ±        1  J-        J- 

So  that  a3  X  a^  X  ft"^  =  «•    And  a"Xa"  ....n  times  =  a. 

247.  It  follows  from  this  plan  of  notation,  that 

a2  X  o^  =  a^  ~^^  •      For  a^  +^  _  ^^i  q^.  a, 

JL  JL  JL  JL    I    -L    I    JL 

a'  xa'  Xa'  =a'~^^~T~'  =a\  &c. 
where  the  multiplication  is  performed  in  the  same  manner 
as  the  multiplication  of  powers,  (Art.  233,)  tliat  is,  by  adding 
the  indices. 

248.  Every  root  as  well  as  every  power  of  1  is  1.  (Art. 
209.)  For  a  root  is  a  factor,  which  multiplied  into  itself  will 
produce  the  given  quantity.  But  no  factor  except  1  can  pro- 
duce 1,  by  being  multiplied  into  itself. 

So  that  1",  1,  V^  V^  ^c-  are  all  equal. 

249.  Negative  indices  are  used  in  the  notation  of  roots,  as 
well  as  of  powers.     See  Art.  207. 

1  1  1  . 

Thus -r=a-i     —=a~t     -j=a-t 

a2  a'  a" 


102  ALGEBRA. 


POWERS  OF  ROOTS. 


250.  It  has  been  shown  in  what  manner  any  power  or 
root  may  be  expressed  by  means  of  an  index.  The  index 
of  a  power  is  a  whole  nnmber.  That  of  a  root  is  a  fraction 
whose  numerator  is  1.  There  is  also  another  class  of  quan- 
tities which  may  be  considered,  either  as  powers  of  roots, 
or  roots  of  powers. 

Suppose  a-  is  multiplied  into  itself,  so  as  to  be  repeated 
three  times  as  a  factor. 

The  product  a2+2-ri  qj.  ^fi   (^Xi^.  247,)  is  evidently  the 

cube  of  a^,  that  is,  the  cube  of  the  square  root  of  a.  This 
fractional  index  denotes,  therefore,  a  power  of  a  root.  The 
denominator  expresses  tlie  root,  and  the  numerator  the  power. 
The  denominator  shows  into  how  many  equal  factors  or  roots 
the  given  quantity  is  resolved  ;  and  the  numeiator  shows  how 
many  of  these  roots  are  to  be  multiplied  together. 

Thus  a^  is  the  4th  power  of  the  cube  root  of  a. 

The  denominator  shows  that  a  is  resolved  into  the  three 

-L  1  -L 

factors  or  roots  a^,  and  a^,  and  a^.  And  the  numerator  shows 
that  four  of  these  are  to  be  multiplied  together ;  which  will 

produce  the  fourth  power  of  a^ ;  that  is, 

X  ±  X  X  A. 

251.  As  a^is  a  power  of  a  root,  so  it  is  a  root  of  a  poioer. 
Let  a  be  raised  to  the  third  power  a^.     The  square  root  of 

3 

this  is  a  .  For  the  root  of  a^  is  a  quantity  which  multiplied 
into  itself  will  produce  a^. 

But  according  to  Art.  247,  a^^a^X^^Xa  ;  and  this 
multiplied  int<J  itself,  (Art.  103,)  is 

a^  Xa^  Xa^  X^^Xa   Xa  — a'- 

3 

Therefore  or  is  the  square  root  of  the  cube  of  a. 

m 

In  the  same  manner,  it  may  be  shown  that  oT  is  the  mth 
power  of  the  nth  root  of  a;  or  the  nth  root  of  the  mth  pow- 


RADICAL  QUANTITIES.  lOS 

er  :  that  is,  a  root  of  a  power  is  equal  to  the  same  power  of  the 
same  root.  For  instance,  the  fourth  power  of  the  cube  root  of 
a,  is  the  same  as  the  cube  root  of  the  fourth  power  of  a. 

252.  Roots,  as  well  as  powers,  of  the  same  letter,  may  be 
multiplied  by  adding  their  exponents.  (Art.  247.)  It  will  be 
easy  to  see,  that  the  same  principle  may  be  extended  to  pow- 
ers of  roots,  when  the  exponents  have  a  common  denomi- 
nator. 

-2.  2.  -2.    i    2.  A 

Thus  a'  xa'  =a''~^'  =a\ 

For  the  first  numerator  shows  how  often  a''  is  taken  as  afac- 
tor  to  produce  a\     (Art.  250.) 

And  the  second  numerator  shows  how  often  a'  is  taken  as 
a  factor  to  produce  a'. 

The  sum  of  the  numerators  therefore,  shows  how  often  the 
root  must  be  taken,  for  the  product.     (Art.  103.) 

2.  X  X 

Or  thus,  a"^  =  a''  XO''^ ' 

2.         X  X         ± 

And         a'  =  «^X«'X«'- 

2.2.XXXXX5 

Therefore  a'  x«'  =«^  X«'  X^t'  X«'  X^'  =a  . 

253.  The  value  of  a  quantity  is  not  altered,  by  applying 
to  it  a  fractional  index  whose  numerator  and  denominator 
are  equal. 

-2-  -3.  ii 

Thus  a =a^=:a^=  a".  For  the  denominator  shows  that 
a  is  resolved  into  a  certam  number  of  factors ;  and  the  nu- 

n 

merator  shows  that  all  these  factors  are  included  in  a». 

3.  X         X  X 

Thus  a^  =a^  xa^  X^^^j  which  is  eoual  to  a. 

^  X  X  X 

And  a'*  =  a"Xa"X(f''''---nih-nes. 

On  the  other  hand,  when  the  numerator  of  a  fractional 
index  becomes  equal  to  the  denominator,  the  expression  may 
be  rendered  more  simple  by  rejecting  the  index. 

n 

Instead  of  a"^,  we  may  write  a. 

254.  The  index  of  a  power  or  root  may  be  exchanged,  for 

any  other  index  of  the  same  value. 

#  ■*■ 

Instead  of  a  ,  we  may  put  a". 


104  ALGEBRA. 

For  in  the  latter  of  these  expressions,  a  is  supposed  to  be 
resolved  into  twice  as  many  factors  as  in  the  former ;  and  the 
numerator  shows  that  twice  as  many  of  these  factors  are  to  be 
multiphed  togetlier.     So  that  the  whole  value  is  not  altered. 

Thus  x^=x^=x\  &c.  that  is,  the  square  of  the  cube  root 
is  the  same,  as  the  fourth  power  of  the  sixth  root,  the  sixth 
power  of  the  ninth  root,  &:c. 

So  a^=a'^=a^  =  ar^.  For  the  value  of  each  of  these  in- 
dices is  2.     (Art.  135.) 

255.  From  the  preceding  article,  it  will  be  easily  seen, 
that  a  fractional  index  may  be  expressed  in  decimals. 

1 .  Thus  a^  =  a'  ^,  or  aP-^ ;  that  is,  the  square  root  is  equal  to 
the  5th  power  of  the  tenth  root. 

i  25 

2.  a*  =  a^~^,  or  a"'^^;  that  is,  the  fourth  root  is  equal  to 
the  25th  power  of  the  1 00th  root. 

3.  J=a''*  5.  J  =  a}-' 

4.  a^=a''  6.  0^  =  0'''" 

In  many  cases,  however,  the  decimal  can  be  only  an  ap- 
proximation to  the  true  index. 

Thus  a^=a'^-^  nearly.  a^  =  a°  •  ^ ^  ^ ^ "  very  nearly. 

In  this  manner,  the  approximation  may  be  carried  to  any 
degree  of  exactness  which  is  required. 

Thus  a^=a' -'''''.  a^=a''''"*\ 

These  decimal  indices  form  a  very  important  class  of  num- 
bers, called  logarithms. 

It  is  frequently  convenient  to  vary  the  notation  of  powers 
of  roots,  by  making  use  of  a  vinculum,  or  the  radical  sign  \/. 
In  doing  this,  we  must  keep  in  mind,  that  the  power  of  a 
root  is  the  same  as  the  root  of  a  power ;  (Art.  251,)  and  also, 
that  the  denominator  of  a  fractional  exponent  expresses  a 
root,  and  the  numerator  a  power.      (Art.  250.) 

i  -Li 

Instead,  therefore,  of  «%  we  may  write  (a^)%  or  (a')  ,  or 


EVOLUTION.  105 

The  first  of  these  three  forms  denotes  the  square  of  the 
cube  root  of  a;  and  each  of  the  two  last,  the  cube  root  of  the 
square  of  a. 

«    ~±r      ~^\\       

So  fln  =:a"'     =a  \  =J^a"'. 

And   (6i-)^"  =(6V)i=:*76V! 

EVOLUTION. 

257.  Evolution  is  the  opposite  of  involution.  One  is  find- 
ing a  power  of  a  quantity,  by  multiplying  it  into  itself.  The 
otli^r  is  finding  a  root,  by  resolving  a  quantity  into  equal  fac- 
tors. A  quantity  is  resolved  into  any  number  of  equal  fac- 
tors, by  dividing  its  index  into  as  many  equal  parts  ;  (Art. 
245.) 

Evolution  may  be  perfomied,  then,  by  the  following  gen- 
eral rule  ; 

Divide  the  index  of  the  quantity  by  the  number 

EXPRESSING  the  ROOT  TO  BE  FOUND. 

Or,  place  over  the  quantity  the  radical  sign  belonging  to 
the  required  root. 

1.  Thus  the  cube  root  of   a^  is  al      For  rt'xa^Xa^=a*. 

Here  6,  the  index  of  the  given  quantity,  is  divided  by  3, 

the  nmiiber  expressing  the  cube  root. 

J. 

2.  The  cube  root  of  aor  a\  is  a^  or  \/a. 

For  a^  xa'  X«%  or  \/aX\/^X\/"=a.  (Arts.  243,  246.) 

3.  The  5th  root  of  ah,  is  {ah)^  or  ^ab. 

4.  The  nth  root  of  a^  is  a  "  or  ^a^. 


5.  The  7th  root  of  U -x,\s  {U-x)''  oxl/U- x. 

6.  The  5th  root  of  a  -  x\,  is  a-xy  or  '^^a  -  a:|  . 

7.  The  cube  root  of  a-,  is  a\    (Art.  163.) 

8.  The  4th  root  of  a-'\s  a'"^' 

9.  The  cube  root  of  a^  is  a^ . 

10.  The  nth  root  of  xT,  is  arn. 

14 


IOC  ALGEBRA. 

258.  According  to  the  rule  just  given,  the  cube  root  of  the 
square  root  is  found,  by  dividing  the  index  J  by  3,  as  in  ex- 
ample 7tli.  But  instead  of  dividing  by  3,  we  may  multiply 
by  i.     For  l^S=h'iri=h  Xh   (Art.  162.) 

So  ~-^n=—X—      Therefore   the  with  root  of   the  nth 
m  m    n 

J.  X  -L 
root  of  a  is  equal  to  a"     '". 

J. 

Thatis,  a"l   =a"^"'  =  a"'\ 

Here  the  two  fractional  indices  are  reduced  to  one  by  mul- 
tiplication. 

It  is  sometimes  necessary  to  reverse  this  process  ;  to  resolve 
an  index  into  two  factors. 


Thus  x^=x^^^^  =x'\  That  is,  the  8th  root  of  x  is  equal 
to  the  square  root  of  the  4th  root. 

Soa+6j     =a+6|  =a-]-b\      . 

It  may  be  necessary  to  observe,  that  resolving  the  index 
into  factors,  is  not  the  same  as  resolving  the  quantity  into 
factors.  The  latter  is  effected,  by  dividing  the  index  into 
parts. 

259.  The  rule  in  Art.  257,  may  be  applied  to  every  case 
in  evolution.  But  when  the  quantity  whose  root  is  to  be 
found,  is  composed  of  several  factors,  there  will  frequently 
be  an  advantage  in  taking  the  root  of  each  of  the  factors 
separately. 

This  is  done  upon  the  principle  that  the  root  of  the  product 
of  several  factors,  is  equal  to  the  product  of  their  roots. 

Thus  \/ab  —  A^a  X  \^h.     For  each  member  of  the  equation 

if  involved,  will  give  tlie  same  power. 
The  square  of  A^/ah  is  ah.  (Art.  241.) 
Thesquareof /\/rtXV^»^sV^*XV^*XV^XV^-(Ai'^-102.) 
But/v/«xV«  =  «-   (Art.  241.)  AndV^XV^=^- 
Therefore  the  square  of  \/^XV^=V^XV^XV^XV^ 

=  a6,  which  is  also  the  square  of  \fah. 

On  the  same  principle,  (a^)"  =a"b". 


EVOLUTION.  107 

"When,  therefore,  a  quantity  consists  of  several  factors,  we 
may  either  extract  the  root  of  the  whole  together;  or  we  may 
fiixl  the  root  of  the  factors  separatel)',  and  then  mukiply  them 
into  each  other. 

Ex.  1.  The  cube  root  of  xy,  is  either  (xy)^  or  x^y^. 

2.  The  5th  root  of  Sy,  is  \/Sy  or  \/SX\/y- 

3.  The  6th  root  of  abh,  is  {abhy,  or  a^b^h^. 

4.  The  cube  root  of  8b,  is  (86)  %  or  2b^, 

5.  The  nth  root  of  x^y^  is  (x"i/)"  or  xy". 

260.  The  root  of  a  fraction  is  equal  to  the  root 

OF    THE    numerator    DIVIDED    BY    THE    ROOT    OF    THE  DENO- 
MINATOR. 

1.  Thus  the  square  root  of  _= —     For — x — =— 

-L  JL  JL 

2.  So  the  nth  i:oot  of-=:^.    For'i^x  — -.n  times  =f. 

b     ^^  ^i     ^^  b 

X         \/x  /^^      Vfl'^ 

3.  The  square  root  of  — ,  is  -^L^.    4.  V  IT^a/^/ 

ay        As/ ay  ^^      ^""^ 

261.  For  determining  what  ns,n  to  prefix  to  a  root,  it  is 
important  to  observe,  that 

An  odd  root  of  any  quantity  has  the  same  sign  as 
the  quantity  itself. 

An  EVEN  ROOT  OF  AN  AFFIRMATIVE  QUANTITY  IS  AM- 
BIGUOUS. 

An  EVEN  ROOT  OF  A  NEGATIVE    QUANTITY    IS    IMPOSSIBLE. 

That  the  3d,  5th,  7th,  or  any  other  odd  root  of  a  quantity 
must  have  the  same  sign  as  the  quantity  itself,  is  evident 
from  Art.  219. 

262.  But  an  even  root  of  an  affirmative  quantity  may  be 
either  affirmative  or  negative.  For,  the  quantity  may  be 
produced  from  the  one,  as  well  as  from  the  other.  (Art.  219.) 

Thus  the  square  root  of  a^  is  -\-a  or  -a. 


108  ALGEBRA. 

An  even  root  of  an  affirmative  quantity  m,  therefore,  said 
to  be  ambiguous,  and  is  marked  with  both  -j-  and  -. 

Thus  the  square  root  of  36,  is  1/^/36. 

The  4th  root  of  x,  is  ±x^. 

The  ambiguity  does  not  exist,  however,  when,  from  the 
nature  of  the  case,  or  a  previous  muUiphcation,  it  is  known 
whether  the  power  has  actually  been  produced  from  a  posi- 
tive or  from  a  negative  quantity.     See  Art.  299. 

263.  But  no  even  root  of  a  negative  quantity  can  be  found. 
The  square  root  cf-a*^  is  neither  -f-^  i^or  -a. 
For  -|-ax4-a  =+«^     And  -ax  -a=-\-a'^  also. 

An  even  root  of  a  negative  quantity  Is,  therefore,  said  to  be 
impossible  or  imaginary. 

There  are  purposes  to  be  answered,  however,  by  applying 
the  radical  sign  to  negative  quantities.  The  expression 
^  -a  is  often  to  be  found  in  algebraic  processes.  For,  al- 
though we  are  unable  to  assign  it  a  rank,  among  either  posi- 
tive or  negative  quantities  ;  yet  we  know  that  when  multi- 
plied into  itself,  its  product  is  -  a,  because  >>/  -  a  is  by  notation 
a  root  of  -  a,  that  is,  a  quantity  which  multiphed  into  itself 
produces  -a. 

This  may,  at  first  view,  seem  to  be  an  exception  to  the 
general  rule  that  the  product  of  two  negatives  is  affirm- 
ative. But  it  is  to  be  considered,  that  ^  -a  is  not  itself  a 
negative  quantity,  but  the  root  of  a  negative  quantity. 

The  mark  of  subtraction  here,  must  not  be  confounded 
with  that  which  is  prefixed  to  the  radical  sign.  The  expres- 
sion A^  —  a  is  not  equivalent  to  -/>/«.  The  former  is  a  root 
of -a;  but  the  latter  is  a  root  of-|-a: 

For  -/>/ax  -^az=i^aa  —  a. 

The  root  of  -  a,  however,  may  be  ambiguous.  It  may  be 
either  -\-a^  -a,ox  -^  -a. 

One  of  the  uses  of  imaginary  expressions  is  to  indicate 
an  impossible  or  absurd  supposition  in  the  statement  of  a 
problem.  Suppose  it  be  required  to  divide  the  nmuber  14 
mto  two  such  parts,  that  their  product  shall  be  60.  If  one 
of  the  parts  be  x,  the  other  will  be  14  -:r.  And  by  the  sup- 
position, 

a:x(14-a;)  =  60,  or  14a: -a:'=:60. 


EVOLUTION  109 

This  reduced,  by  the  rules  in  the  following  section,  will 
give  x=7±^~U. 

As  the  value  of  x  is  here  found  to  contain  an  imaginary 
expression,  we  infer  that  there  is  an  inconsistency  in  the 
statement  of  the  problem:  that  the  number  14  cannot  be 
divided  into  any  two  parts  whose  product  shall  be  60.* 

264.  The  methods  of  extracting  the  roots  of  compound 
quanthies  are  to  be  considered  in  a  future  section.  But 
there  is  one  class  of  these,  the  squares  of  binomial  and  re- 
sidual quantities,  which  it  will  be  proper  to  attend  to  in  this 
place.  It  has  been  shown  (Art.  214,)  that  the  square  of  a 
binomial  quantity  consists  of  three  terms,  two  of  which  are 
complete  powers,  and  the  other  is  a  double  product  of  the 
roots  of  these  powers.     The  square  of  a-{-b,  for  instance,  is 

a:^j^2ab-\-b\ 
two  terms  of  which,  a^  and  U\  are  complete  powers,  and  2ab 
is  twice  the  product  of  a  into  6,  that  is,  the  root  of  a^  into  the 
root  of  b'\ 

Whenever,  therefore,  we  meet  with  a  quantity  of  this  de- 
scription, we  may  know  that  its  square  root  is  a  binomial ; 
and  this  may  be  found,  liy  taking  the  root  of  the  two  terms 
which  are  complete  powers,  and  connecting  them  by  the 
sign  -\-.  The  other  term  disappears  in  the  root.  Thus,  to 
find  the  square  root  of 

take  the  root  of  x%  and  the  root  of  1/^  and  connect  them  by 
the  sign  -f .     The  binomial  root  will  then  be  x-{-y. 

In  a  residual  quantity,  the  double  product  has  the  sign  - 
prefixed,  instead  of  +.  ^  The  square  of  a-b,  for  instance,  is^ 
a''-2ab-\-b\  (Art.  214.)  And  to  obtain  the  root  of  a  quantity 
of  this  description,  we  have  only  to  take  the  roots  of  the  two 
complete  powers,  and  connect  them  by  the  sign  -.  Thus  the 
scjuare  root  of  x^  -2xy-\-y^  is  x  -y.     Hence, 

265.  To   EXTRACT  A  BINOMIAL  OR  RESIDUAL  SQUARE  ROOT, 
TAKE   THE    ROOTS   OF  THE   TWO  TERMS  WHICH  ARE  COMPLETS. 
POWERS,  AND  CONNECT  THEM  BY  THE   SIGN  AVHICH  IS   PREFIX 
ED  TO  THE  OTHER  TERM. 

Ex.  1.  To  find  the  root  of  x''-[-2x-\-\. 

The  two  terms  which  are  complete  powers  are  x"^  and  1 
The  roots  are  x  and  1.      (Art.  248.) 
The  binomial:  root  is,  therefore,  a:-f  1. 


♦  See  Note  F. 


110  ALGEBRA. 

2.  The  square  root  of  0^-2x^1,  isx-\.    (Art.  214.) 

3.  The  square  root  of  a^+a+J,  is  a+J.    (Art.  224.) 

4.  The  square  root  of  a^-^ta-\-^,  is  a-f-f- 

b'  b 

5.  The  square  root  of  a'^-f-at-j- j,  is  «+9' 

o    r^,  2a6     b'  b 

6.  The  square  root  of  «--j- — -f-;^?  ^^  ^-hZ' 

c       cr  c 

266.  A  ROOT  WHOSE  VALUE  CANNOT  BE  EXACTLY  EXPRESS- 
ED IN  NUMBERS,  IS  CALLED  A  SURD. 

Thus  /y/2  is  a  surd,  because  the  square  root  of  2  cannot  be 
expressed  in  numbers,  with  perfect  exactness. 
In  decimals,  it  is  1.41421356  nearly. 

But  though  we  are  unable  to  assign  the  value  of  such  a 
quantity  lohen  taken  alone,  yet  by  multiplying  it  into  itself,  or 
by  combining  it  with  other  quantities,  we  may  produce  ex- 
pressions whose  value  can  be  determined.  There  is,  there- 
fore, a  system  of  rules  generally  appropriated  to  surds.  But 
as  all  quantities  whatever,  when  under  the  same  radical  sign, 
or  having  the  same  index,  may  be  treated  in  nearly  the  same 
manner ;  it  will  be  most  convenient  to  consider  them  toge- 
ther, under  the  general  name  of  Radical  Quantities ;  under- 
standing by  this  term,  every  quantity  which  is  found  under 
a  radical  sign,  or  which  has  a  fractional  index. 

267.  Every  quantity  which  is  not  a  surd,  is  said  to  be 
rational.  But  for  the  purpose  of  distinguishing  between  ra- 
dicals and  other  quantities,  the  term  rational  will  be  applied, 
in  this  section,  to  those  only  which  do  not  appear  under  a 
radical  sign,  and  which  have  not  a  fractional  index. 

REDUCTION  OF  RADICAL  aUANTITIES. 

268.  Before  entering  on  the  consideration  of  the  rules  for 
the  addition,  subtraction,  multiplication  and  division  of  radi- 
cal quantities,  it  will  be  necessary  to  attend  to  the  methods 
of  reducing  them  from  one  form  to  another. 

First,  to  reduce  a  rational  quantity  to  the  form  of  a  radi- 
cal; 

Raise  the  quantity  to  a  power  of  the  same  name  as 

THE    given    root,    AND    THEN     APPLY     THE     CORRESPONDING 
radical    sign    OR    INDEX. 


RADICAL  QUANTITIES.  1 1 1 

Ex.   1 .  Reduce  a  to  the  form  of  the  nth  root. 

The  nth  power  of  a  is  a".     (Art.  211,) 

Over  this,  place  the  radical  sign,  and  it  becomes  \/a*. 

It  is  thus  reduced  to  the  form  of  a  radical  quantity,  with- 
out  any  alteration  of  its  value.     For  \yaP=a'^=za. 

2.  Reduce  4  to  the  form  of  the  cube  root. 

Ans.  \/64  or  (64)  ^ 

3.  Reduce  3a  to  the  form  of  the  4th  root. 

Ans.  ^S\a\ 

4.  Reduce  ^ah  to  the  form  of  the  square  root. 

Ans.  {Wh'^y. 

5.  Reduce  3x«  -  ^  to  the  form  of  the  cube  root. 

Ans,  ^27x«^'.     See  Art.  212. 

6.  Reduce  a^  to  the  form  of  the  cube  root. 
The  cube  of  a'  is  a\     (Art.  220.) 

And  the  cube  root  of  a'  is  Ija^ z=a^\^ . 

In  cases  of  this  kind,  where  a  power  is  to  be  reduced  to 
the  form  of  the  nth  root,  it  must  be  raised  to  the  nth  power, 
not  of  the  given  letter,  but  of  the  power  of  the  letter. 

Thus  in  the  example,  a^  is  the  cube,  not  of  a,  but  of  a'. 

7.  Reduce  a^h^  to  the  form  of  the  square  root. 

8.  Reduce  aJ^  to  the  form  of  the  nth  root. 

269.  Secondly,  to  reduce  quantities  which  have  different 
indices,  to  others  of  the  same  value  having  a  common  index  ; 

1.  Reduce  the  indices  to  a  common  denominator. 

2.  Involve  each  quantity  to  the  power  expressed  by  the 
numerator  of  its  reduced  index. 

3.  Take  the  root  denoted  by  the  common  denominator. 

JL  JL 

Ex.  1.  Reduce  a*  and  6 Mo  a  common  index. 

1st.  The  indices  \  and  i  reduced  to  a  common  denomina- 
tor, are  1^2  and  ^2-      (Art.  146.) 

2d.  The  quantities  a  and  h  involved  to  the  powers  express- 
ed by  the  two  numerators,  are  a^  and  h^. 


113  ALGEBRA. 

Sd.  The  root  denoted  by  the  common  denominator  is  jS. 

Tlie  answer,  tlien,  is  a^|'^  and  b'^\^^. 
The  two  quantities  are  thus  reduced  to  a  common  index, 
without  any  aUeration  in  their  values. 

For  by  Art.  254,  a^=a'^\  which  by  Art.  258,  ^¥\^. 
And  universally  a"  =  a"'"  =a'"|'"''. 

_L  Si. 

2.  Reduce  a^  and  bx^  to  a  common  index. 
The   indices   reduced  to  a  common    denominator  are   } 
and  f. 


The  quantities  then,  are  a^  and  (bx)^,  or  a^j*',  and  b*x^\^ 

3.  Reduce  a^  and  6 ".     Ans.  a^"\"  and  6". 

JL  ±  X  -L 

4.  Reduce  x"  and  y".     Ans.  x'^l"'"  And  y"\'"", 

5.  Reduce  2^  and  st      Ans.  8^  and  9^. 


6.  Reduce  (a+t)^ and  (a;- j/)^  Ans.  a-|-6  |    snidx-y  \  • 

XX  Si  X 

7.  Reduce  a^  and  6^.         8.   Reduce  x^  and  5^. 

270.  When  it  is  required  to  reduce,  a  quantity  to  a  given 
index ; 

Divide  the  index  of  the  quantity  by  the  given  index,  place 
the  quotient  over  the  quantity,  and  set  the  given  index  over 
the  whole. 

This  is  merely  resolving  the  original  index  into  twc  factors, 
according  to  Art.  258. 

Ex.  1.  Reduce  a^  to  the  index  J. 

By  Art.  162,  i-i-i  =  ixf=i=h 
This  is  the  index  to  be  placed  over  a,  which  then  becomes 

JL  .  ~T|^ 

a^  ;  and  the  given  index  set  over  this,  makes  it  a^j  ,  the  an- 
swer. 

3. 

2.  Reduce  a^  and  a:^  to  the  common  index  h 

2-^1=  2  X  3  =  6,  the  first  index  > 

ii-T-3"=tx3=f,  the  second  index      > 

Therefore  (a")^  and  {x^)^  are  the  quantities  required. 


RADICAL  QUANTITIES.  113 

3.  Reduce  4^  and  3^,  to  the  common  index    . 

Answer,   (4^)\nd  (3^)^. 

271.  Thirdly,  to  remove  a  part  of  a  root  from  under  the 
radical  sign  ; 

If  the  quantity  can  be  resolved  into  two  factors,  one  of 
which  is  an  exact  power  of  the  same  name  with  the  root ; 

FIND     THE    ROOT    OF    THIS    POWER,    AND    PREFIX    IT    TO    THE 
OTHER    FACTOR,  WITH    THE    RADICAL    SIGN  BETWEEN    THEM. 

This  rule  is  founded  on  the  principle,  that  the  root  of  the 
product  of  two  factors  is  equal  to  the  product  of  their  roots. 
(Art.  259.) 

It  will  generally  be  best  to  resolve  the  radical  quantity  into 
such  factors,  that  one  of  them  shall  be  the  greatest  power 
which  will  divide  the  quantity  without  a  remainder.  If 
there  is  no  exact  power  v/hich  will  divide  the  quantity,  the 
reduction  cannot  be  made. 

Ex.   1.  Remove  a  factor  from  \/S. 

The  greatest  square  wiiich  will  divide  8  is  4. 
We  may  then  resolve  8  into  the  factors  4  and  2.  For  4x2  =  8. 

The  root  of  this  product  is  equal  to  the  product  of  the  roots 
of  its  factors  ;  that  is,  \/S=\/4xV^- 

But  \/4=:2.  Instead  of  \/4,  therefore,  we  may  substitute 
its  equal  2.     We  then  have  2  x  V^  or  2^/2. 

This  is  commonly  called  reducing  a  radical  quantity  to  its 
most  simple  terms.  But  the  learner  may  not  perhaps  at  once 
perceive,  that  2\/2  is  a  more  simple  expression  than  >\/8. 

2.  Reduce  A^d'x.     Ans.  \/a^X\/x=ax\/x=aA^x. 

3.  Reduce  ^/\S.     Ans.  V9x2  =  V^ X V^  =  3 V2. 

4.  Reduce  \/646^.     Ans  {/64b'X\/c=4b^c. 

'    /TFh  a'    /  b 

5.  Reduce    N/  ^/-      Ans.   cV  c5*     (Art.  260.) 

6.  Reduce  J^a^b.      Ans.  a^b,  or  rt6". 

7.  Reduce    (a^-a-b)^.     Ans.  a(a-b)^. 

8.  Reduce  {oia%y.     Ans.  Sa%2by. 

9.  Reduce  >^98aFx.  10.  Reduce  \/d'-\-a''b\ 

15 


114  ALGEBRA. 

272.  By  a  contrary  process,  the  co-efficient  of  a  radical 
quantity  may  be  introduced  under  the  radical  sign. 

1.  Thus,  a-^6=^a"6. 

For  a^z^a^or  an.      (Art.  253.)    And  ^a'•X^/^  =  \/^"^• 
Here  the  co-efficient  a  is  first  raised  to  a  power  of  the  same 
name  as  the  radical  part,  and  is  then  introduced  as  a  factor 
under  the  radical  sign. 

2.  a{x-by  =  {a'x^'^y=^{a'x-a'by. 

3.  2ab{2aby  =  {\6a'b')t 


4.    Z 


a  I    ¥c    U      /    a^b'^c    \i 


hW-{-b 


ADDITION  AND  SUBTRACTION  OF  RADICAL 
QUANTITIES. 

273.  Radical  quantities  may  be  added  like  rational  quan- 
tities, by  loriting  them  one  after  another  with  their  signs,  (Art. 
69.) 

Thus  the  sum  of  \/a  and  \/b,  is  \/a-\-\/b. 

And  the  sum  of  a^  - h^  and x*  -y" ,  is  a^  - h^-^x*  - y" . 

But  in  many  cases,  several  terms  may  be  reduced  to  one, 
as  in  Arts.  72  and  74. 

The  sum  of  2\/a  and  3\/ais  2A^a-\-S\/a=5\/a. 
For  it  is  evident  that  twice  the  root  of  a,  and  three  times 
the  root  of  a,  are  five  times  the  root  of  a.    Hence, 

274.  Wlienthe  quantities  to  be  added  have  the  same  radi- 
cal part,  under  the  same  radical  sign  or  index  ;  add  the  ra- 
tional parts,  and  to  the  sum  annex  the  radical  parts. 

If  no  rational  quantity  is  prefixed  to  the  radical  sign,  1  is 
always  to  be  understood.     (Art.  244.) 


To     2;:/ay 
Add     :{/ay 

5\/a 
-2V« 

3(:r+/0^ 
4{x-\-hy 

7{x-^hy 

5bh'              ax/b-h 
7bh'             yA^b-h 

Sum  SJ{/ay 

{a-\-y)X\/b-h 

RADICAL  QUANTITIES.  115 

275.  If  the  radical  parts  are  originally  different,  they  may 
sometimes  be  made  alike,  by  the  reductions  in  the  preceding 
articles. 

1 .  Add  \/8  to  /s/oO.  Here  the  radical  parts  are  not  the 
same.  But  bv  the  reduction  in  Art,  271,  /\/8=2\/2,  and 
V50=5V2.    "^The  sum  then  is  7V2. 

2.  Add  A^XQh  to  \fAh.     Ans.  A\fh-\-2\^h  =  Q\/h. 

3.  Add  V^""-^  to  V^'^-  Ans.  aVa:+fe\/^=(M-^^)XV* 

4.  Add  (36a^y)^  to  (25i/)  ^     Ans.  (6a4-5)  x/- 

5.  Add  VI  8a  to  ^\/2a. 

276.  But  if  the  radical  parts,  after  redaction,  are  different 
or  have  different  exponents,  they  cannot  be  united  in  the 
same  term;  and  must  be  added  by  writing  them  one  after  the 
other. 

The  sum  of  Sy^  ^"tnii  1\/a,  is  '^\/h-\-2\/a. 

It  is  manifest  that  three  times  tlie  root  of  6,  and  twice  the 
root  of  a,  are  neither  five  times  the  root  of  6,  nor  five  times 
the  root  of  a,  unless  h  and  a  are  equal. 

The  sum  of  Xja  and  Xja,  is  \/a-\-\/a. 

The  square  root  of  a,  and  the  cube  root  of  a,  are  neither 
twice  the  square  root,  nor  twice  the  cube  root  of  a. 

277.  Subtraction  of  radical  quantities  is  to  be  performed  in 
the  same  manner  as  addition,  except  that  the  signs  in  the  sub- 
trahend are  to  be  changed  according  to  Art.  82. 

J.  .  i 

From       Ai/ay  4\/a-^x  Sh^  a{x-]~y)     -a  " 

Sub.       S\^ay  S\/a-{-x  -  5h^  b{x-{-y)  -2a-" 


Diff.  -2\/ay  8h/ 


a  " 


From  V50,  subtract  V8.     Ans.  5\/2  -  2V2  =  3 \/2.  (Art. 
275.) 

From  ^6  *y,  subtract   ^by*.    Ans.  {b-y)x  {/by. 
From  J^x,  subtract  ^x. 

MULTIPLICATION  OF  RADICAL  QUANTITIES. 

278.   Radical   quantities   may   be  multiplied,  like   other 


116  ALGEBRA. 

quantities,  by  writing  the  factors  one  after  another,  either 
with  or  without  the  sign  of  multipHcation  between  them. 
(Art.  93.) 

Thus  the  product  of  \fa  into  \/^j  is  \/"XV^- 

X  -L  JL    J. 

The  product  of  h^  into  i/^  is  /l^2/'^ 

But  it  is  often  expedient  to  bring  the  factors  under  the 
same  radical  sign.  This  may  be  done,  if  they  are  first  re- 
duced to  a  common  index. 

Thus  \/a:  X  Vl/ —  V^-  ^^^  ^^^®  ^^^^  ^^  ^^^  product  of 
several  factors  is  equal  to  the  product  of  their  roots.  (Art. 
259.)     Hence, 

279.  Quantities  under  the  same  radical  sign  or  in- 
dex, MAY  BE  MULTIPLIED  TOGETHER  LIKE  RATIONAL  QUAN- 
TITIES, THE  PRODUCT  BEING  PLACED  UNDER  THE  COMMON 
RADICAL    SIGN  OR  INDEX.* 

Multiply  \/x  into  ^?/,  that  is,  x^  into  \f . 
The  quantities  reduced  to  the  same  index,  (Art.  269.)  are 
(a:')S  and  (i/^)®  and  their  product  is,  {x^'^'^y  =%/3^^ . 

Mult. 
Into 

Prod. 


Multiply     Js/^xh\\\io\/%xh.     Prod.    A^X^x'^h'' z=.^xh. 
In  this  manner  the  product  of  radical  quantities  often  be- 
comes rational. 

Thus  the  product  of  \/2  into  \/\8=a^S6  =  6. 

And  the  product  of  (a^y^)^  into  (a'Uj)*  =  (a*y*)*=ay. 

280.  Roots  of  the  same  letter  or  quantity  may  be 
multiplied,  by  adding  their  fractional  exponents. 

The  exponents,  like  all  other  fractions,  must  be  reduced 
to  a  common  denominator,  before  they  can  be  united  in  one 
term.     (Art.  148.) 

*  The  case  of  an  imaginary  root  of  a  negative  quantity  may  be  considered 
an  exception.     (Art.  263.) 


\/a-|-w 
A^a-m 

A^dx 
^/hy 

a^          (a-\-yr 

xi      (b+hy 

(a^xr 

± 
a"' 

-L 

x" 

1 

s/a^  -  m^ 

gjn^m)«.« 

RADICAL  QUANTITIES.  117 


U„i_J+i_> 


6  +  6  «6 


Thus  a2xa'=a^   '=«       =tt 
The  values  of  the  roots  are  not  altered,  by  reducing  their 
indices  to  a  common  denominator.   (Art.  254.) 

Therefore  the  first  factor  cfi  =  a''  ( 
And  the  second  a'^=a^  ) 

Buta^=a'xcf''X(^-   (Art.  250.) 

J2.  X  ± 

And  a^  =  a^  X«^ 
-L       -L       J.       J.       1       a. 
The  product  therefore  is  a®  x^*^  X«®  X«^  Xa°=«  • 

And  in  all  instances  of  this  nature,  the  common  denomin- 
ator of  the  indices  denotes  a  certain  root ;  and  the  sum  of 
the  numerators,  shows  how  often  this  is  to  be  repeated  as  a 
factor  to  produce  the  required  product. 


X-* 


Thus 

a"Xa"'=«'"''X 

n             m+n 

Mult. 

3,i 

a^X'i* 

(a+b)i 

{a. 

-y) 

Into 

f 

a* 

(a+b)' 

(«- 

■y) 

U  ...  -3. 


Prod.       Sy'^  («+^)'  a:~Ti 

The  product  of  y^  into  i/~Ms  y^~^z=y^. 

-L  _i  J._JL 

The  product  of  a"  into  a    ",  is  a"    "  =  a°=\. 

The  product  of  a^  into  a^  =a^  x^^^  =a  • 

281.  From  the  last  example  it  will  be  seen,  that  powers 
and  roots  may  be  multiplied  by  a  common  rule.  This  is  one 
of  the  many  advantages  derived  from  the  notation  by  frac- 
tional indices.  Any  quantities  whatever  may  be  reduced  to 
the  form  of  radicals,  (Art.  268,)  and  may  then  be  subjected 
to  the  same  modes  of  operation. 

Thus  y^xy^=y   ^=y  ^  • 

±  14-J-         "±L 

And  xx^""  =x    "  =  x^  . 


118  ALGEBRA. 

The  product  will  become  rational,  whenever  the  numera- 
tor of  the  index  can  be  exactly  divided  by  the  denominator. 

JL  JJ.  JL-2. 

Thus  a^ X «^  X «^  =a^  =  a. 

And  {a+hyx  (a+b)  '^  ={a-^hy  =  a-\-b. 

i      A      4. 
And  a^  x«^  =«""  =  «• 

282.  When  radical  quantities  which  are  reduced  to  the 
same  index,  have  rational  co-efficients,  the  rational 

PARTS     MAY     BE     MULTIPLIED    TOGETHER,    AND    THEIR    PRO- 
DUCT PREFIXED    TO    THE    PRODUCT    OF  THE    RADICAL    PARTS. 

1.  Multiply  a\/b  into  c\/d. 

The  product  of  the  rational  parts  is  ac. 
The  product  of  the  radical  parts  is  \/bd. 
And  the  whole  product  is  ac\/bd. 
For  aA^b  is  axV^-   i-^^'^-  ^44.)  And  c.\/d  is  cx  V^- 

By  Art.  102,  axV^  "^^^  cX\^d,  is  «XV^XcXV^i  ^^ 
by  changing  the  order  of  the  factors, 

«  X  c  X  V^  X  V^ = ^^  X  V^^^ = ^^  V^^ 

2.  Multiply  ax^  into  bd^ . 

When  the  radical  parts  are  reduced  to  a  common  index, 

JL  JL 

the  factors  become  a[x^y  and  b{d^y . 

The  product  then  is  a6(a;V^)^ 

But  in  cases  of  this  nature  we  may  save  the  trouble  of  re- 
ducing to  a  common  index,  by  multiplying  as  in  Art.  278. 

1    .  J.    .  IX 

Thus  ax^  into  bd^  is  ax-bd^ . 

Jl  _  -L 

Mult.  a{b-{-xy         a^/y^         a\fx  ax     ^     x\/^ 

Into  y{b~xy          b\fh\j.      b\/x  61/ "'     y\/^ 

Prod.         ay{b''-xy  ab\/x^=abx  Sxy 

283.  If  the  rational  quantities,  instead  of  being  co-efficients 
to  the  radical  quantities,  are  connected  with  them  by  the 
signs  -j-  and  -  ,  each  term  in  the  multfipUer  must  be  multi- 
plied into  each  in  the  multiplicand,  as  in  Art.  100. 


RADICAL  QUANTITIES.  119 


Multiply  a-fV^ 
Into  c-\-\/d 

ac-{-c\/b 

a\/d-\-\/bd 

ac-\-c\/b-\^a\/d-{-\/bd. 
The  product  o(    a-\-y\/y  into  l-\-r ^/^J  is 

1.  Multiply  \/a  into  ^b.  Ans.  ^a^, 

2.  Multiply  5\^5  into  3\/8.       Ans.  30\/io. 

3.  Multiply  2V3  into  3^/4.       Ans.  6^432. 

4.  Multiply  V^/ into  ^a6.         Ans.    ^^cfbW. 

5.  Multiply  ^^  into  ^?^.    Ans.      /3^. 

6.  Multiply  a(a  -  a:)^  into  (c  -  rf)  x  (aa:)^. 

Ans.  {ac  -  ad)  xi^^J^-  ax^)^. 

DIVISION  OF  RADICAL  QUANTITIES. 

284.  The  division  of  radical  quantities  may  be  expressed, 
by  writing  the  divisor  under  the  dividend,  in  the  form  of  a 
fraction. 

Thus  the  quotient  of  \/a  divided  by  /s/b,  is  ^^. 

And  (a+h)^  divided  by  {b-\-xf  is  i±thL^ 

In  these  instances,  the  radical  sign  or  index  is  separately 
applied  to  the  numerator  and  the  denominator.  But  if  the 
divisor  and  dividend  are  reduced  to  the  same  index  or  radical 
sign,  this  may  be  applied  to  the  whole  quotient. 

Thus  .r/a-i-^6=  v^=  "  /t     For  the  root  of  a  fraction 

is  equal  to  the  root  of  the  numerator  divided  by  the  root  of 
the  denominator.   (Art.  260.) 


120  ALGEBRA. 

Again,  /y/a6-r->Y/6  =  yy/a.     For  the  product  of  this  quotient 
into  the  divisor  is  equal  to  the  dividend,  tltat  is, 

j:i/aX\/b  =  ^ab.     Hence, 

285.  Quantities  under  the  same  radical  sign  or  index 

MAY  be  divided  LIKE  RATIONAL  QUANTITIES,  THE  QUOTIENT 
BEING  PLACED  UNDER  THE  COMMON  RADICAL  SIGN  OR  INDEX. 

Divide  {xYY  by  y^. 


J.  J. 

These  reduced  to  the  same  index  are  (x^)  ^  and  (t/^)  ®  : 

_L  i  1 

And  the  quotient  is  {x^)  ^  =zx^  =x^. 
Divide  V6a^    \/dk?     (a'-{-axy     (n'li)^     (^V)^ 


By        \/Sx 

A/dx 

J 

{axr 

(«!/)* 

Quot.    V2«' 

(a'+x)^ 

{«!/)*• 

286.  A  ROOT  IS  DIVIDED  BY  ANOTHER  ROOT  OF  THE 
SAME  LETTER  OR  QUANTITY,  BY  SUBTRACTING  THE  INDEX 
OF  THE  DIVISOR  FROM  THAT  OF  THE  DIVIDEND. 

Thus  a^-^a'=a^     '  =  a'~'=a'=a\ 

For  a^=a^=za^  X^"  X^^  and  this  divided  by  a'  is 

«^\y«6^«6  .1         JL  ^  J. 

a' 
In  the  same  manner,  it  may  be  shown  that  a'"-i-a "  =  a"*     ". 
Divide   (3a)  "^         {axf         a"^         (b+y)^  (r'y')' 


Quot.    (3a) i  a"  (r'f)"^ 

Powers  and  roots  may  be  brought  promiscuously  together, 
and  divided  according  to  the  same  rule.     See  Art.  281. 


RADICAL  QUx\NTITIES.  121 

Thus  a'^J=a''-^=a^.       For  a^xa^  =  «^=«'. 


So  y^-T-y^^y" 


287.  When  radical  quantities  which  are  reduced  to  the 
same  index  have  rational  co-efficients,  the  rationai 

PARTS    MAY    BE    DIVIDED    SEPARATELY,  AND  THEIR  QUOTIENT 
PREFIXED  TO  THE   QUOTIENT   OF  THE  RADICAL   PARTS. 

Thus  ac\/bd-^a\^b  =  CA^d.     For  tliis  quotient  muUiphed 
into  the  divisor  is  equal  to  the  dividend. 

Divide       24x\/ay     \Sdh\/bx     by{a'x''Y      16V32      b^xy 

By  6  V«         27tVc       yid^)"         ^V-i        Vy 


4x\/y  b{a''x)"  b\^x 


Divide  abi^x^y  by  a  (a:)t 


These  reduced  to  tlie  same  index  are  ab{x"b)^  and  a^x'-y. 

The  quotient  then  is  6(6)^=  (6^) \      (Art.  272.) 

To  save  the  trouble  of  reducing  to  a  common  index,  the 
division  may  be  expressed  in  the  form  of  a  fraction. 

JL 

The  quotient  will  then  be  ^—^ . 

a{xY 


1. 

Divide  'it\/bc  by  3V«c. 

Alls. 

e      /6' 

2. 

Divide  10 VI 08  by  5\/L 

Ans. 

2\/21^Q. 

3. 

Divide  10V27  by  2V3. 

Ans. 

15. 

4. 

Divide  8VlOSby2V6. 

Ans. 

12V2. 

5. 

Divide  {a'b'd'Y  hyd\ 

Ans. 

(abY. 

6. 

Divide  (16a'  -  V2a''xY  by  2a. 

Ans. 

(4a-3a:)^. 

INVOLUTION  OF  RADICAL  QUANTITIES. 

288.  Radical  quantities,  like  powers,  are  involved 

BY  multiplying  THE  INDEX  OF  THE  ROOT  INTO  THE  INDEX  OF 
THE  REQUIRED  POWER. 

16 


122  ALGEBRA. 

1.  The  square  of  J'=c^^'^=^.     For  a^  xo'  =  a'. 

2.  Thecubeof«^=a'^^^  =  a\  Yox  J'xa^Xa^  =  ^' 

n 

3.  And  universally,  the  nth  power  of  a^'^a""^    =a  . 

J.  J-  -L 

For  the  ?ith  power  of  a"*  =  a"* X «'" n  times,  and  the  sum 

of  the  indices  will  then  be  ^. 

4.  The  5th  power  of  a^  y\  is  a^t/^     Or,  by  reducing  the 
roots  to  a  common  index, 

(ay)*X^=(aV)* 


5.  The  cube  of  a"a;%  is  a  "a;'"  or  (a"'a;")" 

J2.   i    .       ^  4- 

6.  The  square  of  a^x'^,  is  a^a;' 


The  cube  of  a'  is«'^^^ 


«^  =  «. 


J. 


And  the  nth  power  of  a",  is  an  =  a.     That  is, 

289.       A  ROOT  IS  RAISED  TO  A  POWER  OF  THE  SAME  NAME, 
BY  REMOVING  THE  INDEX  OR  RADICAL  SIGN. 


Thus  tne  cube  of  \/b-\-x,  is  b-]-x. 

And  the  nth  power  of  (ci-y)"',  is  (a  -  y.) 

290.  When  the  radical  quantities  have  rational  co-efficients, 
these  must  also  be  involved. 

1.  The  square  of  a\/x,  is  a^J^/^^- 
For  ay/xX(t'\/^=^^ V^^ ' 

2.  The  nth  power  of  a'^x"",  is  a"""  x"".  * 


3.  The  square  of  aA^x - y,  is  a'^xi^-y-) 

4.  The  cube  of  Sa{/y,  is  27ahj. 

291.  But  if  the  radical  quantities  are  connected  with 
others  by  the  signs  +  and -,  they  must  be  involved  by  a 
multiplication  of  the  several  terms,  as  in  Art.  213. 


RADICAL  QUANTITIES.  123 

Ex.  1 .  Required  the  squares  of  a-\-\fy  and  a  -  \/y. 

aJrVy  (^-Vy 


a^+ ^  V?/  ^^  -  (^\^y 

(iVy+y  -  aVy+y 


2.  Required  the  cube  of  a  -  A/b. 

3.  Required  the  cube  of  2d-\-\^x. 


292.  It  is  unnecessary  to  give  a  separate  rule  for  the  evo- 
lution of  radical  quantities,  that  is,  for  finding  the  root  of  a 
quantity  which  is  ah'eady  a  root.  The  operation  is  the  same 
as  in  other  cases  of  evolution.  Tlie  fractional  index  of  the 
radical  quantity  is  to  be  divided,  by  the  number  expressing 
the  root  to  be  found.  Or,  the  radical  sign  belonging  to  the 
required  root,  may  be  placed  over  the  given  quantity.  (Art. 
237.)  If  there  are  rational  co-efiicients,  the  roots  of  these 
must  also  be  extracted. 

J         i_^o       ± 

Thus,  the  square  root  of  a"^,  is  a^  •  "^^a^ 

The  cube  root  of  a{xy)'^,  is  a^{xy)^. 


V  «^ 


The  nth  root  of  a\/by,  is  >/  a\/hy. 

293.  It  may  be  proper  to  observe,  that  dividing  the  frac- 
tional index  of  a  root  is  the  same  in  effect,  as  multiplying  the 
number  which  is  placed  over  the  radical  sign.  For  this 
number  corresponds  with  the  denominator  of  the  fractional 
index ;  and  a  fraction  is  divided,  by  multiplying  its  denomi- 
nator 

Thus  V«=a*-  V«=at 

V«=«'  ''\/a=a^\ 

On  the  other  hand,  multiplying  the  fractional  index  is 
equivalent  to  dividing  the  number  which  is  placed  over  the 
radical  sign. 

Thus  the  square  of  X/a  or  a%  is  \/a  or  a^=a^. 


124  ALGEBRA. 

293.  b.  In  algebraic  calculations,  we  have  sometimes 
occasion  to  seek  for  a  factor,  which  multiplied  into  a  given 
radical  quantity,  will  render  the  product  rational.  In  the 
case  of  a  simple  radical,  such  a  factor  is  easily  found.  For 
if  the  nth  root  of  any  quantity,  be  multiplied  by  the  same 
root  raised  to  a  power  whose  index  is  n-  1,  the  product  will 
be  the  given  quantity. 

Thus  \/x X  V'^'""'  ^^  ^'   X ^  "  =x''  =x. 

±  n-l 

n  ^ 

And  (x^y)    X{x-\-y)       =x-\-y. 

So  \/axV"="-     -^^^  %yax\/(f^=\/(^^=(^' 

And  {/ax{/a'=a,  &c.     And  {a-{~by  x{a-\-b)^  =  a-{-b. 

And  {x-]-yyx(x-\-y)*  =  x-{-y. 

293.  c.  A  factor  which  will  produce  a  rational  product, 
when  multiplied  into  a  binomial  surd  containing  only  the 
square  root,  may  be  found  by  applying  the  principle,  that 
the  product  of  the  sum  and  difference  of  tAvo  quantities,  is 
equal  to  the  difference  of  their  squares.  (Art.  235.)  The 
binomial  itself,  after  the  sign  which  connects  the  terms  is 
changed  from -{-  to-,  or  hom-to-j-,  will  be  the  factor 
required. 

Thus  (V«+V^)  X  (V«  -  V^)  =  V^'  -  \/b'  =  a-  b,  Avhich 
is  free  from  radicals. 

So(i4-V2)x(i-V2)  =  i-2=-i. 

And  (3  -  2V2)  X  (3+ W^)  =  ^- 

When  the  compound  surd  consists  of  more  than  two  terms, 
it  may  be  reduced,  by  successive  multiplications,  first  to  a 
binomial  surd,  and  then  to  a  rational  quantity. 

Thus  (V10-V2-V3)  X(\/10+V2+V3)  =  5-2V6, 
a  binomial  surd. 

And  (5  -  2 V6)  X  (5+^6)  =  1  • 
Therefore  (\/\0  -  \/2  -  ^^)  multiphed  into  (V10+V2+ 
V3)X(5+2V6)  =  1. 

293.  d.  It  is  sometimes  desirable  to  clear  from  radical  signs 
the  numerator  or  denominator  of  a  fraction.  This  may  be 
effected,  without  altering  the  value  of  the  fraction,  if  the 


RADICAL  QUANTITIES.  125 

numerator  and  denominator  be  both  multiplied  by  a  factor 
which  will  render  either  of  them  rational,  as  the  case  may 
require. 

1.  If  both  parts  of  the  fraction  !^_.be  multiplied  by  \/a, 

it  will  become  ^ — XV^—    ^^^  [^^  which  the  numerator  is  a 

rational  quantity. 

Or  if  both  parts  of  the  given  fraction  be  multiplied  by  ,\/x, 

it  will  become  V_^^  [^^  which  the  denominator  is  rational. 

X 

2.  The  fraction  J_Jx{a+^)^^h^X(a+^)^ 

{a+xY        (a+r)*+*  "+'' 

3.  Thefrac.io«^^=(y-+ili:t!=J±^. 


4.  The  fraction  4  =  -fiJ--_=^V^_. 
x"      x"Xi'~^  ^ 

5.  The  fraction      V^     _   V2x(3+V2)     _2+3V2 

3-V2      (3-V2)(3+V2)  7 

6.  The  fraction ? = giV^+V-J =  AJb 

V5-V2     (V5-V2)(V5+V2) 

6_6x5J_6 

7.  The  fraction    i"    114  — T"^ 

54     54-1-4      o 

8.  The  fraction 

8  _  8x(V3-V2-l)(-V2)  _4_ 


V3+V2+1      (V3+V2+1)(V3-V2-1)(-V2) 

2y6+2V2. 

o 

9.  Reduce to  a  fraction  having  a  rational  denominator. 

V3 

10.  Reduce  ^~\^_  to  a  fraction  having  a  rational  denom- 

a-\-\/h 
inator. 

293.  e.  The  arithmetical  operation  of  finding  the  proximate 
value  of  a  fractional  surd,  may  be  shortened,  by  rendering 


126  ALGEBRA. 


• 


either  the  numerator  or  the  denominator  rational.  The  root 
of  a  fraction  is  equal  to  the  root  of  the  numerator  divided  by 
the  root  of  the  denominator.      (Art.  260.) 

Thus  «    /--^.  But  this  may  be  reduced  to. 


V 


or  VaXVfc"~\     (^j-t   293.  d.) 

The  square  root  of  ?  is  ^  or  _^,  or  "S^. 
b       \/h        \/ab  b 

When  the  fraction  is  thrown  into  this  form,  the  process  of 
extracting-  the  root  arithmetically,  will  be  confined  either  to 
the  numerator,  or  to  the  denominator. 

Thus  the  square  root  of  -=^-^V^  ^V2=vS. 

Examples  for  practice. 

1.  Find  the  4th  root  of  81a^ 

2.  Find  the  6th  root  of  (a+6)  -\ 

3.  Find  the  nth  root  of  {x  -y)\ 

4.  Find  the  cube  root  of  -  125a V. 

5.  Find  the  square  root  of  — —. 

*  9xY 

6.  Find  the  5th  root  of  ^^^'^'°. 

243 

7.  Find  the  square  root  of  x"^  -  6bx-{-9b'^ 

8.  Find  the  square  root  of  a^-\-ay-\-^. 

9.  Reduce  ax^  to  the  form  of  the  6th  root. 

10.  Reduce  -3t/  to  the  form  of  the  cube  root. 

11.  Reduce  a^  and  a^  to  a  common  index. 

12.  Reduce  4^  and  5^  to  a  common  index. 

JL  JL  .J. 

13.  Reduce  a'^  and  6*  to  the  common  index^. 

14.  Reduce  2^  and  4*  to  the  common  index'. 


RADICAL  QUANTITIES.  127 


15.  Remove  a  factor  from  \/294. 


16.  Remove  a  factor  from  ^/x^-d^x^. 


17.  Find  the  smn  and  difference  of  a^/I^o^x  and  \^A(ix. 

18.  Find  the  sum  and  difference  of  ^192  and  ^24. 

19.  Multiply  7\/T8  into  5\/4. 

20.  Multiply  4+2V2  into  2-a^2. 

21.  Multiply  a{a-\-\fcY  into  h{a-\^c)\, 

22.  Multiply  2(a+fe)'' into  3(a4-6)^ 

23.  Divide  6V54  by  3V2. 

24.  Divide  4^72  by  2^18. 

25.  Divide  v^  by  Xjl. 

26.  Divide8V5l2"by4\/2. 

27.  Find  the  cube  of  \l\/¥\. 

28.  Find  the  square  of  5+ V^- 

29.  Find  the  4th  power  of  iv^. 

30.  Find  the  cube  of  \fx-\/h. 

31.  Find  a  factor  which  will  make  Xyy  rational. 

32.  Find  a  factor  which  will  make  ^5  -  \/x  rational. 

33.  Reduce  ^l_^  to  a  fraction  having  a  rational  numerator. 

\/^ 

34.  Reduce  — ^ to  a  fraction  having  a  rational  de- 

nominator. 


128  ALGEBRA. 


SECTION  X. 


REDUCTION  OF  EQUATIONS  BY  INVOLUTION 
AND  EVOLUTION. 

Art.  294.  IN  an  equation,  the  letter  which  expresses  the 
unknown  quantity  is  sometimes  found  under  a  radical  sign. 
We  may  have  ^x—a. 

To  clear  this  of  the  radical  sign,  let  each  member  of  the 
equation  be  squared,  that  is,  multiplied  into  itself.  We  shall 
then  have 

^xy^^^x^aa.  Or,  (Art.  289,)  x=c^. 

The  equality  of  the  sides  is  not  affected  by  this  operation, 
because  each  is  only  multiplied  into  itself,  that  is,  equal  quan- 
tities are  multiplied  into  equal  quantities. 

The  same  principle 'is  applicable  to  any  root  whatever. — 
If  \/x=a ;  then  x=(£'.  For  by  Art.  289,  a  root  is  raised  to 
a  power  of  the  same  name,  by  removing  the  index  or  radical 
sign.     Hence, 

295.  When  the  unknown  quantity  is  under  a  radical 

SIGN,  the  equation  IS  REDUCED  BY  INVOLVING  BOTH  SIDES, 

to  a  power  of  the  same  name,  as  the  root  expressed  by  the 
radical  sign. 

It  will  generally  be  expedient  to  make  the  necessary  trans- 
positions, hefore  involving  the  quantities ;  so  that  all  those 
which  are  not  under  the  radical  sign  may  stand  on  one  side 
of  the  equation. 

Ex.  1.  Reduce  the  equation  -\/a:-|-4  =  9 

Transposing -|- 4  .\/x—^-\  —  b. 

Involving  both  sides  a?  =  5^= 25. 

.  Reduce  the  equation  a-\-\/x-hz=zd 

By  transposition,  \/x—  d-\-h  -  a 

By  involution,  x=i  {d-\-h  ~  a)". 


EQUATIONS. 


IS9 


3.  Reduce  the  equation 
Involving  both  sides, 
And 

4.  Reduce  the  equation 
Clearmg  of  fractions. 
And 

Involving  both  sides, 
And 

5.  Reduce  the  equation 


a+1  =4^=64 
x=63. 


4-f3V^-4=64-J 
8+6V^-4=l3 


_     3-frf 


MuUi plying  by  \/a'^-\-^yxy 
And 
Involving  both  sides, 


«'4-V^=3+fi 
^/x=S~\-d-a' 
x={S-ird-a'')\ 


In  the  first  step  in  this  example,  multiplying  the  first  mem- 
ber into  yyyc^-j-\/x,  that  is,  into  itself,  is  the  same  as  squar- 
ing it,  which  is  done  by  taking  away  its  radical  sign.  The 
other  member  being  a  fraction,  is  multiplied  into  a  quantity- 
equal  to  its  denominator,  by  cancelling  the  denominator. 
(Art.  159.)  There  remains  a  radical  sign  over  x,  which 
must  be  removed  by  involving  both  sides  of  the  equation. 

6.  Reduce  3+2 V^-i= 6. 

7.  Reduce  4^- =  8. 

8.  Reduce  (2a;-f  3)^+4=7. 

9.  Reduce  ,yy\2-{-x=2-\-\/x. 
10.  Reduce  \/a:- a  =  \/a;-i/\/a. 


11.  Reduce  V^XV^+ 2=2+ >y/5j:. 


12.  Reduce  ir_^z=  V^f 

^yx  X 

13.  Reduce  V^±!?=V^ 

\/x-{-4       ^x-\-6 


38 


Ans. 

x=iH. 

Ans, 

x=20. 

Ans. 

x=\2. 

Ans. 

x=4. 

Ans. 

2oa 

Is* 

Ans.  x=  _-. 
2C 

Ans. 

1 
^-1-a 

Ans. 

x=4. 

17 


130  ALGEBRA. 

14.  Reduce  \/-^-{-y\/a-|-x= — '^  Ans.  a;=ia. 

\fa-\-x 

2a^ 

15.  Reduce  x^AJd^-\-x'^=—p^ Ans.  x^aVi- 


16.  Reduce  a;-|-rt=:/\/^^+^"\/6'^4-a;^'    Ans.  a'; 

4  o 

17.  Reduce  \^/2-^x^\/x=  .    Ans.   a:=-. 


4a 


18.  Reduce  Va:- 32=16 -V^-  Ans.  rr=81. 

19.  Reduce  aJI^A1z=.2\^x^\.        Ans.  a;=16. 

a/6^-2     4V6^-9 

20.  Reduce     , —  r~ci=  a    , —  i  ^-       Ans.  rr=6. 

REDUCTION  OF  EQUATIONS  BY  EVOLUTION. 

296.  In  many  equations,  the  letter  which  expresses  the 
unknown  quantity  is  involved  to  some  power.  Thus  in  the 
equation 

n"2=16 
we  have  the  vahie  of  the  square  of  x,  but  not  of  x  itself.     If 
the  square  root  of  both  sides  be  extracted,  we  shall  have 

x=4. 
The  equality  of  the  members  is  not  afTected  by  this  reduc- 
tion.    For  if  two  quantities  or  sets  of  quantities  are  equal, 
their  roots  are  also  equal. 

If  (x-\-a)"=:h^h,  ihenx-{-a=J!^b-]-h.     Hence, 

297.  When  the  expression  containing  the  unknown 
quantity  is  a  power,  the  equation  is  reduced  by  ex- 
TRACTING THE  root  of  both  SIDES,  a  root  of  the  same  name 
as  the  powder. 

Ex.  1.  Reduce  the  equation  6-\-x^  -8=7 

By  transposition  a:^=7-}-8  -  6  =  9 

By  evolution         '  x=±a^9=±S. 

The  signs  -\-  and  -  are  both  placed  before  \/9,  because 
an  even  root  of  an  affirmative  quantity  is  ambiguous.  (Art. 
261.) 


EQUATIONS.  131 

2.  Reduce  the  equation  5x^-S0  =  a^-X-S4. 
Transposing,  &c.  a;*  =16 

By  evolution,  x=±4. 

2  2 

3.  Reduce  the  equation,  a-j-^  =  /i-^ 

b  d 

Clearing  of  fractions,  &c.  ^.^^bdh-abd 

b-\-d 
By  evolution,  ^__-f /fcrf/,-aferf\i. 

4.  Reduce  the  equation,  a-\-djf=\0  -  x" 
Transposing,  &c.  a"  =  ^  ^  ~  ^ 


By  evolution. 


dJ\-\ 
/10-«\i 


298.  From  the  preceding  articles,  it  will  be  easy  to  see  in 
what  manner  an  equation  is  to  be  reduced,  when  the  ex- 
pression containing  the  unknown  quantity  is  a  power,  and  at 
the  same  time  under  a  radical  sign  ;  that  is,  when  it  is  a  root 
of  a  power.  Both  involution  and  evolution  will  be  necessary 
in  thjs  case. 

Ex.  1.  Reduce  the  equation  ^ar^=4. 

By  involution  ar^=4^=64 

By  evolution  a:=±\/64=±8. 

2.  Reduce  the  equation  V^  -a=h-d 

By  involution  x^-a=h^-  2hd-{-d^ 

And  ar  =  h?-  <^hd-{-d'-{-a 

By  evolution  x=\/h^-  'ilhd-^d^-\-a. 

3.  Reduce  the  equation  {x-\-a)  ^  =  —L 

{x-a)^ 

Multiplying  by  (re -a)^    (Art.  279.)  (x' -a=^)i  =  a+6 
By  involution  a:^  -  a^ = a^_j_2a6+6'^ 

Trans,  and  uniting  terms     x^=2a^~\-2ab-\-b^ 

By  evolution  x=  r2a''-|-2a6+6')i 


132  ALGEBRA. 

Problems. 

Prob.  I.  A  gentleman  being  asked  his  age,  replied,  "  If 
you  add  to  it  ten  years,  and  extract  the  square  root  of  the 
sum,  and  from  this  root  subtract  2,  the  remainder  will  be  6," 
What  was  his  age  1 


By  the  conditions  of  the  problem  ,^a:-}- 10  -2  =  6 

By  transposition,  \/x-\-lO=:6-\-2  =  S 

By  involution,  rr+10=8'=64. 

And  a:=64- 10=54. 


Proof  (Art.  194.)  v'54+10-2  =  6. 

Prob.  2.  If  to  a  certain  number  22577  be  added,  and  the 
Hquare  root  of  the  sum  be  extracted,  and  from  this  163  be 
subtracted,  the  remainder  will  be  237.  What  is  the  num- 
ber] 

Let  x=  the  number  sought.  6  =  163 

a=22577  c=237. 

By  the  conditions  proposed  \/x-\-a-b=c 

By  transposition,  \/x-\-a=c-{-b 

By  involution,  x-\-a=  {c-\-by 

And  x=(c-^by-a 

Restoring  the  numbers,  (Art.  52.)  x=(237+163)3- 22577 
That  is  ar=  160000 -22577=  137423. 


Proof      V137423+22577  - 163  =  237. 

299.  When  an  equation  is  reduced  by  extracting  an  even 
root  of  a  quantity,  the  solution  does  not  determine  whether 
the  answer  is  positive  or  negative.  (Art.  297.)  But  what 
is  thus  left  ambiguous  by  the  algebraic  process,  is  frequently 
settled  by  the  statement  of  the  problem. 

Prob.  3.  A  merchant  gains  in  trade  a  sum,  to  which  320 
dollars  bears  the  same  proportion  as  five  times  this  sum  does 
to  2500.     What  is  the  amount  gained  ? 

Let  ar=the  sum  required. 
a=320. 
6  =  2500. 


EQUATIONS.  133 

By  the  supposition  a:x  ::  5x  :b 

Multiplying  the  extremes  and  means        5x^  =  ab 

-{if 

Restoring  the  numbers,  x=  /ggg^^^QQji  ^ 400. 

Here  the  answer  is  not  marked  as  ambiguous,  because  by 
the  statement  of  the  problem  it  is  gain,  and  not  loss,  li 
must  therefore  be  positive.  This  might  be  determined,  in 
the  present  instance,  even  from  the  algebraic  process. 
Whenever  the  root  of  x^  is  ambiguous,  it  is  because  we  are 
ignorant  whether  the  powder  has  been  produced  by  the  mul- 
tiplication of  -\-x,  or  of -a:,  into  itself.  (Art.  262.)  But 
here  we  have  the  multiplication  actually  performed.  By 
turning  back  to  the  two  first  steps  of  the  equation,  Ave  find 
that  5x^  was  produced  by  multiplying  5x  into  x,  that  is  -]-5x 
into  -j-ar. 

Prob.  4.  The  distance  to  a  certain  place  is  such,  that  if 
96  be  subtracted  from  the  square  of  the  number  of  miles,  the 
remainder  will  be  48.     What  is  the  distance  ? 

Let  x=  the  distance  required. 
By  the  supposition  x^  -0Q  =  48 

Therefore  x=\/lU=l2. 

Prob.  5.  If  three  times  the  square  of  a  certain  number  be 
divided  by  four,  and  if  the  quotient  be  diminished  by  12,  the 
remainder  will  be  180.     What  is  the  number  1 

By  the  supposition  —  -12  =  1 80. 

Therefore  a;=V256=16. 

"Prob.  6.  What  number  is  that,  the  fourth  part  of  whose 
square  being  subtracted  from  8,  leaves  a  remainder  equal  to 
four  1  Ans.  4. 

Prob.  7.  What  two  numbers  are  those,  whose  sum  is  to  the 
greater  as  10  to  7 ;  and  w^hose  sum  multiphed  into  the  les? 
produces  270  1 

Let  10a^= their  sum. 

Then  7a; = the  greater,  and  3a:  =  the  less. 

Therefore  a:  =  3,  and  the  numbers  required  are  21  and  9 


134  ALGEBRA. 

Prob.  8.  What  two  numbers  are  those,  whose  difference  is 
to  the  greater  as  2  :  9,  and  the  difference  of  whose  squares 
is  1281  Ans.  18  and  14. 

Prob.  9.  It  is  required  to  divide  tlie  number  18  into  two 
such  parts,  tliat  the  squares  of  those  parts  may  be  to  each 
other  as  25  to  16. 

Let  x=  the  greater  part.         Then  18  -  a:r=the  less. 

By  the  condition  proposed  a;^ :  (18  -  a;)^ :  :  25  :  16. 

Therefore  1 6a;^=  25  X  ( 1 8  -  .t)^ 

By  evolution  4x = 5  x  ( 1 8  -  .r. ) 
And  rrrzlO. 

Prob.  10.  It  is  required  to  divide  the  number  14  into  two 
such  parts,  that  the  quotient  of  the  greater  divided  by  the 
less,  may  be  to  the  quotient  of  the  less  divided  by  the  greater, 
as  16:9.  Ans.  The  parts  are  8  and  6. 

Prob.  11.  What  two  numbers  are  as  5  to  4,  the  sum  of 
whose  cubes  is  5103  1 

Let  5a:  and  4a; = the  two  numbers. 

Then  x=S,  and  the  numbers  are  15  and  12. 

Prob.  12.  Two  travellers  A  and  B  set  out  to  meet  each 
other,  A  leaving  the  town  C,  at  the  same  time  that  B  left  D. 
They  travelled  the  direct  road  between  C  and  D;  and  on 
meeting,  it  appeared  that  A  had  travelled  18  miles  more 
than  B,  and  that  A  could  have  gone  JB's  distance  in  lof  days, 
but  B  w^ould  have  been  28  days  in  going  A''s  distance.  Re- 
quired the  distance  between  C  and  D. 

Let  a:=the  immber  of  miles  A  travelled. 
Then  x-  18= the  number  B  travelled. 

_    1  Q 

=./Z's  daily  progress. 

—  z=  Ks  daily  progress. 

2b 

Therefore  a; :  a;  -  18  :  :  — — :  — 
15}      28 

This  reduced  gives  a: =72,  .^'s  distance. 

The  whole  distance,  therefore,  from  C  to  D=:126  miles. 


EQUATIONS.  135 

Prob.  13.  Find  two  numbers  which  are  to  each  other  as  8 
to  5,  and  whose  product  is  360.  Ans.  24  and  15. 

Prob.  14.  A  gentleman  bought  two  pieces  of  silk,  v/hich 
together  measured  36  yards.  Each  of  them  cost  as  many 
shillings  by  the  yard,  as  there  were  yards  in  the  piece,  and 
their  whole  prices  were  as  4  to  1.  What  were  the  lengths 
of  the  pieces  1  Ans.  24  and  12  yards. 

Prob.  15.  Find  two  numbers  which  are  to  each  other  as 
3  to  2  ;  and  the  difference  of  whose  fourth  powers  is  to  the 
sum  of  their  cubes,  as  26  to  7. 

Ans.     The  numbers  are  6  and  4. 

Prob.  16.  Several  gentlemen  made  an  excursion,  each 
taking  the  same  sum  of  m-oney.  Each  had  as  many  servants 
attending  him  as  there  were  gentlemen  ;  the  number  of  dol- 
lars which  each  had  was  double  the  number  of  all  the  ser- 
vants, and  the  whole  sum  of  money  taken  out  was  3456  dol- 
lars.    How  many  gentlemen  were  there?  Ans.  12. 

Prob.  17.  A  detachment  of  soldiers  from  a  regiment  being 
ordered  to  march  on  a  particular  service,  each  company  fur- 
nished four  times  as  many  men  as  there  were  companies  in 
the  whole  regiment ;  but  these  being  found  insufficient,  each 
company  furnished  three  men  more  ;  when  their  number  was 
found  to  be  increased  in  the  ratio  of  17  to  16.  How  many 
companies  were  there  in  the  regiment  1  Ans.  12. 

AFFECTED  QUADRATIC  EQUATIONS. 

300.  Equations  are  divided  into  classes,  which  are  distin- 
guished from  each  other  by  the  power  of  the  letter  that  ex- 
presses the  unknown  quantity.  Those  which  contain  only 
the  first  power  of  the  unknown  quantity  are  called  equations 
of  one  dimension,  or  equations  of  the  first  degree.  Those  in 
which  the  highest  power  of  the  unknown  quantity  is  a  square, 
are  called  quadratic,  or  equations  of  the  second  degree ; 
those  in  which  the  highest  power  is  a  cube,  ea  nations  of  the 
third  degree,  &c. 

Thus  x=a-\-h,  is  an  equation  of  the  first  degree. 

x^=c,   and  x^-{-ax=d,  are   quadratic  equations,  or 
equations  of  the  second  degree. 

x^=h,  and  x^-^ax'-^bxzzzd,  are  cubic  equations,  or 
nations  of  the  third  degree. 


136  ALGEBRA. 

301.  Equations  are  also  divided  into  pure  and  affected 
equations.  A  pure  equation  contains  only  one  power  of  the 
unknown  quantity.  This  may  be  the  first,  second,  third,  or 
any  other  power.  An  aflfected  equation  contains  different 
powers  of  the  unknown  quantity.     Thus, 

(  x'^=d  -  i,  is  a  pure  quadratic  equation. 
(  x'^-\-bx  =d,  an  affected  quadratic  equation. 
^  a-'— 6-c,  a  pure  cubic  equation. 
I  x^-\-ax~-\-bx=h,  an  affected  cubic  equation. 

A  i)iire  equation  is  also  called  a  simple  equation.  But  this 
t^nn  has  been  apphed  in  too  vague  a  manner.  By  some 
writers,  it  is  extended  to  pure  equations  of  every  degree  ;  by 
others,  it  is  confined  to  those  of  the  first  degree. 

In  a  pure  equation,  all  the  terms  which  contain  the  un- 
known quantity  may  be  united  in  one,  (Art.  185,)  and  the 
equation,  however  complicated  in  other  respects,  may  be  re- 
duced by  the  rules  which  have  already  been  given.  But  in 
an  affected  equation,  as  the  unknown  quantity  is  raised  to  dif- 
ferent powers,  the  terms  containing  these  powders  cannot  be 
united.  (Art.  230.)  There  are  particular  rules  for  the  reduc- 
ticn  of  quadratic,  cubic,  and  biquadratic  equations.  Of  these, 
only  the  first  will  be  considered  at  present. 

302.  An  affected  quadratic  equation  is  one  which 
contains  the  unknown  quantity  in  one  term,  and  the 
square  of  that  quantity  in  another  term. 

The  unknown  quantity  may  be  originally  in  several  terms 
of  the  equation.  But  ali  these  may  be  reduced  to  two,  one 
containing  the  unkilown  quantity,  and  the  other  its  square. 

303.  It  has  already  been  shown  that  a  pure  quadratic  is 
solved  by  extracting  the  root  of  both  sides  of  the  equation.  An 
affected  quadratic  may  be  solved  in  the  same  way,  if  the 
member  which  contains  the  unknown  quantity  is  an  exact 
square.     Thus  tlie  equation 

x''J^2ax-{-a''=b-\-h. 
may  be  reduced  by  evolution.     For  the  fiist  member  is  the 
square  of  a  binomial  quantity.      (Art.  264.)     And  its  root  is 
x-\-  a.     Therefore, 

x~\.a=\/b-\-hy  and  by  transposing  o, 
x  =  \/b--\-h-  a. 


QUADRATIC  EQUATIONS.  137 

304.  But  it  is  not  often  the  case,  that  a  member  of  an  af- 
fected quadratic  equation  is  an  exact  square,  till  an  addi- 
tional term  is  applied,  for  the  purpose  of  making  the  required 
reduction.     In  the  equation 

the  side  containing  the  unknown  quantity  is  not  a  complete 
square.  The  two  terms  of  which  it  is  composed  are  indeed 
such  as  might  belong  to  the  square  of  a  binomial  quantity, 
(^rt.  214.)  But  one  term  is  wanting.  We  have  then  to  in- 
quire, in  what  way  this  may  be  suppUed.  From  having  two 
terms  of  the  square  of  a  binomial  given,  how  shall  we  find 
the  third  ? 

Of  the  three  terms,  two  are  complete  powers,  and  the 
other  is  twice  the  product  of  the  roots  of  these  powers;  (Art. 
214,)  or  which  is  the  same  thing,  the  product  of  one  of  the 
roots  into  twice  the  other.     In  tlie  expression 

x''-\-2ax, 

the  term  2ax  consists  of  the  factors  2a  and  x.  The  latter  is 
the  unknown  quantity.  The  other  factor  2a  may  be  consid- 
ered the  co-effident  of  tlie  unknown  quantity  ;  a  co-efficient 
being  another  name  for  a  factor.  (Art.  41.)  As  x  is  the 
root  of  the  first  term  x" ;  the  other  factor  2a  is  twice  the  root 
of  the  third  term,  which  is  wanted  to  complete  the  square. 
Therefore  half  2a  is  the  root  of  the  deficient  term,  and  a*  is 
the  term  itself.     The  square  completed  is 

a;2_|_2ax+a% 

where  it  will  be  seen  that  the  last  term  a^  is  the  square  of 
half  2a,  and  2a  is  the  co-efl[icient  of  x,  the  root  of  the  first 
term. 

In  the  same  manner,  it  may  be  proved,  that  the  last  term 
of  the  square  of  any  binomial  quantity,  is  equal  to  the  square 
of  half  the  co-efl[icient  of  the  root  of  the  first  term.  From 
this  principle  is  derived  the  following  rule  : 

305.  To  COMPLETE  THE  SQUARE  in  an  affected  quadratic 
equation:  take  the  square  of  half  the  co-efficient  of 

THE  FIRST  POWER  OF  THE  UNKNOWN  QUANTITY,  AND  ADD  IT 
TO  BOTH  SIDES  OF  THE  EQUATION. 

Before  completing  the  square,  the  known  and  unknown 
quantities  must  be  brought  on  opposite  sides  of  the  equation 

18 


138  ALGEBRA. 

by  transposition ;  and  the  highest  power  of  the  unknown 
quantity  must  have  the  affirmative  sign,  and  be  cleared  of 
fractions,  co-efficients,  &c.     See  Arts,  308,  9,  10,  11. 

After  the  square  is  completed,  the  equation  is  reduced,  by 
extracting  the  square  root  of  both  sides,  and  transposing  the 
known  part  of  the  binomial  root.      (Art.  303.) 

The  quantity  which  is  added  to  one  side  of  the  equation, 
to  complete  the  square,  must  be  added  to  the  other  side  also, 
to  preserve  the  equality  of  the  two  members.     (Ax.  1.) 

306.  It  will  be  important  for  the  learner  to  distinguish  be- 
tween what  is  peculiar  in  the  reduction  of  quadratic  equa- 
tions, and  what  is  common  to  this  and  the  other  kinds  which 
have  already  been  considered.  The  peculiar  part,  in  the 
resolution  of  affected  quadratics,  is  the  completing  of  the 
square.  The  other  steps  are  similar  to  those  by  w^liich  pure 
equations  are  reduced. 

For  the  purpose  of  rendering  the  completing  of  the  square 
familiar,  there  will  be  an  advantage  in  beginning  with  exam- 
ples in  which  the  equation  is  already  prepared  for  this  step. 

Ex.  1.  Reduce  the  equation         ar-\-6ax=b 

Completing  the  square,     x''-^6ax-\-9a'^=9a'^-{~b 
Extracting  both  sides  (Art.  303.)    x-]-Sa=±\/9'a^^b 


And  x=^Sa±\/9a^-\-b. 

Here  the  co-efficient  of  x,  in  the  first  step,  is  6ci; 

The  square  of  half  this  is  9a^,  which  being  added  to  both 
sides  completes  the  square.  The  equation  is  then  reduced 
by  extracting  the  root  of  6ach  member,  in  the  same  manner 
as  in  Art.  297,  excepting  that  the  square  here  being  that  of 
a  binomial,  its  root  is  found  by  the  rule  in  Art.  265. 

2.  Reduce  the  equation  x"  ~8bx  —  h 

Completing  the  square,  x"^  -Sbx-\-lQb'^=:\6b^--\-h 

Extracting  both  sides  x  -  4b=±A^*\Gb'^-{-h 

And  x=4btA^lM^Ti. 

In  this  example,  half  the  co-efficient  of  a-  is  46,  the  square 
of  which  166"  is  to  be  added  to  both  sides  of  the  equation. 


QUADRATIC   EQUATIONS.  139 

3.  Reduce  the  equation         x'^-{-ax=b-{-h 

2  0 

Completing  the  square,     x--\-ax-]—  =  —-\~b-]-h 
By  evolution  a;-f-=±(-+^+/i)^ 

And  x=-|±g+6+/t)i 

4.  Reduce  the  equation       x-  -x=h-d 
Completing  the  square,  x"^  -x-\-l  =  {-\-h-d 

And  x=i±{i-{-h-d)K 

Here  the  co-efficient  of  a:  is  1,  the  square  of  half  which  is  J. 

5.  Reduce  the  equation       x^-\-Sx=d-{-6 
Completing  the  square,  x'^-\-Sx-\-i=i-]-d-{-6 
And  x=-f±{i-\-dJr6)K 

6.  Reduce  the  equation  x^  -abx=ab-cd 

Completing  the  square,  x^  -  abx-\-  —  = -f-a6  -  cd 

4        4 

And  x='l^±^^^^+ab-ci\K 

7.  Reduce  the  equation      x'^-\-—=h 

Completing  the  square,  x''-}-—J^—  =-l~^h 

b      46^    4b'^ 


And 


2b   Ub'^  I  ^ 


By  Art.  158,  __=    x^.     The  co-efficient  of  a^,  therefore, 
b       b 

is  |.     Half  of  this  is  ^,  (Art.  163.)  the  square  of  which  is 
46' 


140  ALGEBRA. 


8.  Reduce  tlie  equation        x^--=z7h 

b 


X        \  I 

Completing  the  square,  x"^  -  —I- — =_---j-7/i. 
^        ^  ^        '  b^4b^     4b^^ 

Here  the  fraction  £=:lx.'r.     (Art.   158.)     Therefore   the 
b      b 

co-efficient  of  a;  is  _. 
b 

307.  In  these  and  similar  instances,  the  root  of  the  third 
term  of  the  completed  square  is  easily  found,  because  this 
root  is  the  same  half  co-efficient  from  which  the  term  has 
just  been  derived.     (Art.  304.)     Thus  in  the  last  example, 

half  the  co-efficient  of  a:  is  — ,  and  this  is  the  root  of  the 

2b' 

third  term  — - 

4b' 

308.  When  the  first  power  of  the  unknown  quantity  is  in 
several  terms,  these  should  be  united  in  one,  if  they  can  be 
by  the  rules  for  reduction  in  addition.  But  if  there  are  lite^ 
ral  co-efficients,  these  may  be  considered  as  constituting,  to- 
gether, a  compound  co-efficient  or  factor,  into  which  the  un- 
known quantity  is  multiplied. 

Thus  ax-\-bx-j-dx=(a-^b-\-d)xx-  (Art.  120.)  The 
square  of  half  this  compound  co-efficient  is  to  be  added  to 
both  sides  of  the  equation. 

1.  Reduce  the  equation         x^-{-Sx-]-2x-\-xz=d 
Uniting  terms  x^-{-Gx=d 
Completing  the  square     x^--{-6x-\-9  =  9-\-d 

And  a:=-3J:V9+^- 

2.  Reduce  the  equation         x^-\-ax-\-bx—h 
By  Art.  120.  x^-^(a-\-b)  Xx=h 

Therefore  x'  +  ia+h)  Xx-\-  (^)  '  =  (^)  '+h 
By  evolution  a:4-f^=±     /  (^] '  +h 

And  ^     x=-^^t^]^+^ 


QUADRATIC   EQUATIONS.  141 

3.  Reduce  the  equation  x'-]-ax-x=b 
By  Art.  120    x''-\-{a-l)x^=b 

Therefore  x''-^{a-\)xx-\-  (~y=  l~Y-\-b 

309.  After  becoming  familiar  witli  the  inethod  of  complet-  ^i. 
ing  the  square,  in  affected  quadratic  equations,  it  will  be 
proper  to  attend  to  the  steps  which  are  preparatory  to  this. 
Here,  however,  little  more  is  necessary,  than  an  apj3lication 
of  rules  already  given.  The  knov/n  ond  unknown  quanti- 
ties must  be  brought  on  opposite  sides  of  the  equation  by 
transposition.  And  it  will  generally  be  expedient  to  make 
the  square  of  the  unknown  quantity  the  first  or  leading  term, 
as  in  the  preceding  examples.  This  indeed  is  not  essential. 
But  it  will  show,  to  the  best  advantage,  the  arrangement  of 
the  terms  in  the  completed  square. 

1 .  Reduce  the  equation  a-{-ox  -Sb  =  Sx  -  x^ 
Transp.  and  uniting  terms  x'^-{-2x=zSb -a 
Completing  the  square  x'^-{-2x-\-\  =  'i-l-Sb  -  a 
And                        *  x=z  -  l±^/l-{-^h-a. 

X       3() 

2.  Reduce  the  equation  _= -4 

^  2     a:+2 

Clearing  of  fractions,  &c.  x^-{-l0x=56 

Completing  the  square  x'^-\-10x-\-25  =  2o-\-56  =  Sl 

And  x=-  ot^Sl  =-  5i9. 

310.  If  the  highest  power  of  the  unknown  quantity  has 
any  co-efficient,  or  divisor,  it  must,  before  the  square  is  com- 
pleted, by  the  rule  in  Art.  305,  be  freed  from  these,  by  multi- 
plication or  division,  as  in  Arts.  ISO  and  184. 

1.  Reduce  the  equation  ar-}-24a -  6h='[2x  -  bx^ 

Transp.  and  uniting  terms,  Qx  -  \2x=Qh-  24a 
Dividing  by  6,  x^-  2x  =  h-4a 

Completing  the  square,      x"^  -  2a;-f-l  =  l-j-h  -  ia 
Extracting  and  transp.        x=  l±^l-{-h  -  4a. 


142  ALGEBRA. 

bx* 
2.  Reduce  the  equation  h-\-2x=d-  — 

a 

Clearing  of  fractions  bx'^-]-2ax=ad-  ah 

T>,.   .J.       ,      ,                  2  1  2ax     ad  -  ah 
Dividing  by  b,  x'-\----=  — 

mi       c  2  1  2ax  ,  d^     a^  ,  ad  -  ah 

Therefore  a'+_+_  =  -+_^ 

.     1  a-X-  (a^  .  ad-  ah\\ 

And  .=  -_t(^,+-^-)   . 

311.  If  the  square  of  the  unknown  quantity  is  in  several 
tey-ms,  the  equation  must  be  divided  by  all  the  co-efficients 
of  this  square,  as  in  Art.  185. 

1 .  Reduce  the  equation  bx--\-dx^  -4x=b-h 

Dividing  by  b-ird,  (An.  121.)  x'  -  _i^=^-— ^ 
==    -^     ^  ^       .   bJ^d     b-\-d 

Therefore  -= A+.  /7A)V^-- 

2.  Reduce  the  equation  ax'^-[-x  =  h-^Sx~x* 
Transp.  and  uniting  terms    ax'^-{-x^  -  2x=h 

Dividing  by  a+l,  " 


a-\- 1     a-{- 1 


Comp.  the  square  x"^ - — ^+( -]  =  I r ) 


a-f-1      \ct+l/       \a-\-\/       a-\-l 


Extracting  and  transp.  x=z  — 7 _ /v/  ( 7) 


rt-j-l-'N/    \a-\-lj       a-j-1 
There  is  another  method  of  completing  the  square,  which, 
m  many  cases,  particularly  those  in  which  the  higliest  power 
of  the  unknown  quantity  has  a  co-efficient,  is  more  simple 
in  its  application,  than  that  given  in  Art.  305. 
Let  a3i^-\-bx  =  d. 
If  the  equation  be  multiphed  by  4a,  and  if  b^  be  added  to 
both  sides,  it  will  become 

4aV+4a6a:4-6^=4(wi+&' ; 
the  first  member  of  which  is  a  complete  power  of  2ax-{-b. 
Hence, 

311.  6.  In  a  quadratic  equation,  the  square  may  be  com- 
pleted, by  mviltiplying  the  equation  into  4  times  the  co-effi- 


QUADRATIC   EQUATIONS*  I43 

cient  of  the  highest  power  of  the  unknown  quantity  and  ad- 
ding to  both  sides,  the  square  of  the  co-efficient  of  the  lowest 
power. 

The  advantage  of  this  metliod  is,  that  it  avoids  the  intro- 
duction of  fractions,  in  completing  the  square. 

This  will  be  seen,  by  solving  an  equation  by  both  methods. 

Let  a3^-\-dx=h. 

Completing  the  square  by  the  rule  just  given  ; 

4a^r-|-  4adx-{-  d^ = 4ah-\-  d' 

Extracting  the  root         2ax-\-d=±yy/4ah-fd' 

And  :r=:^V4^5Zl 

2a 


Completing  the  square  of  the  given  equation  by  Arts.  305 

d  310  ;  x'-{-^-{.^,=^l-L^  . 

a      4a      a     4a^ 

Extracting  the  root  a:4-^=+^  /-4-—. 

^2a     "V  a^4a^ 


And 


2a  N/  a     4a^' 


If  a=l,  the  rule  will  be  reduced  to  this:  "Multiply  the 
equation  by  4,  and  add  to  both  sides  the  scfiare  of  the  co- 
efficient of  a:." 

Let  x^-{-dx=^h 
Completing  the  square         4x'^-{-4dx-{-d^=4h-\-d^ 

Extracting  the  root  2x-{-d=z±A^4h-{-dr 

And  .=  Z^V^±?. 

2 

1.  Reduce  the  equation  Sx'^-{-5x=i42 

Completing  the  square  3 6a:''-|-60a;-f  25  =  529 

Therefore  x=3. 

3.  Reduce  the  equation  x"-  I5x=  -54 

Completing  the  square  4x'^  -  60a;-|-225  =  9 

Therefore  2a:=  151:3  =  18  or  12. 

312.  In  the  square  of  a  binomial,  the  first  and  last  terms 
are  always  positive.     For  each  is  the  square  of  one  of  t-he 


144  ALGEBRA. 

terms  of  (he  root.  (Art.  214.)  But  every  square  is  positive. 
(Art  218.)  If  then  -  x^  occurs  in  an  equation,  it  cannot,  with 
this  sign,  form  a  part  of  the  square  of  a  binomial.  But  if 
all  the  signs  in  tlie  equation  be  changed,  the  equahty  of  the 
sides  will  be  preserved,  (Art.  177,)  the  term  -  x^  will  become 
positive,  and  the  square  may  be  completed. 

1.  Reduce  the  equation  -x^-\-'2>x=d-h 

Changing  all  the  signs  x^ -2x  —  h-d 


Therefore  x=\t\f\-\-h-d 

2.  Reduce  the  equation  Ax-x^=z  -\2 

Xwn.  x=2t\^\Q, 

313.  In  a  quadratic  equation,  the  Hrst  term  x^  is  the  square 
of  a  single  letter.  But  a  binomial  quantity  may  consist  of 
terms,  one  Ci*  both  of  which  are  already  powers. 

Thus  x'^~\-a  is  a  binomial,  and  its  square  is 

a;'-'_|-2rtx'+a^ 
where  the  index  of  x  in  the  first  term  is  twice  as  great  as  in 
the  second.  When  the  third  term  is  deficient,  I  he  square 
may  be  completed  in  the  same  manner  as  that  of  any  other 
binomial.  For  the  middle  term  is  twice  the  product  of  the 
roots  of  the  two  others. 

So  the  square  of  a;"-I-«,  is  ar''-{-2aa;'*-|-a^ 

And  the  square  of  a; "-[-«}  is  x''-\-2ax"-\-d^. 
Therefore, 

314.  Any  equation  which  contains  oni.y  two  dif- 
ferent POWERS  OR  roots  OF  THE  UNKNOWN  QUANTITY, 
THE  INDEX  OF  ONE  OF  WHICH  IS  TWICE  THAT  OF  THE 
OTHER,  MAY  BE  RESOLVED  IN  THE  SAME  MANNER  AS  A  QUA- 
DRATIC   EQUATION,    BY    COMPLETING    THE    SQUARE. 

It  must  be  observed,  however,  that  in  the  binomial  root, 
the  letter  expressing  the  unknown  quantity  may  still  have  a 
fractional  or  integral  index,  so  that  a  farther  extraction,  ac- 
cording to  Art.  297,  may  be  necessary. 

1.  Reduce  the  equation  x^-x^=h-  a 

Completing  the  square  a;'  -  x^-\-\z=.\-\-h  —  a 

Extracting  and  transposing  x'^=\t\/^-\-h-a 

Extracting  again,  (Art.  297,)  x=t\/\t^/{\J^h  -  a) 


QUADRATIC  EQUATIONS.  145 

2.  Reduce  the  equation  af"-4bx"=a 


Answer  x=±^2b±\/{4b'^-{-a.) 

3.  Reduce  the  equation  x-\-4^x=h-n 
Completing  the  square  x-\-4^x-\-4=h-  n-\-4 
Extracting  and  transp.  ^yx=  -  2t\/h-n-{-4 
Involving  x=(-  2±/y//i -  n-|-4)'. 

4.  Reduce  the  equation  x"-\-8x"  =za-{-b 

2  ± 

Completing  the  square  x" -^Sx" -\-\6  =  a-\~b-{-\6 

Extracting  and  transp.  x"=  -  4±y\/a~{-b-\-\6 

Involving  x={-  4±\/a-\~b-\-'{6y. 

315.  The  solution  of  a  quadratic  equation,  whether  pure 
or  affected,  gives  tAvo  results.  For  after  the  equation  is  re- 
duced, it  contains  an  amhiguous  root.  In  a  pure  quadratic, 
this  rojot  is  the  whole  value  of  the  unknown  quantity.  (Art. 
2-97.) 

Thus  the  equation  a'~=64 

Becomes,  when  reduced  x=±yy/64. 

That  is,  the  value  of  x  is  either  -f-8  or  -  8,  for  each  of 
these  is  a  root  of  64.  Here  both  the  values  of  x  are  the 
same,  except  that  they  have  contrary  signs.  This  will  be 
the  case  in  every  pure  quadratic  equation,  because  the  whole 
of  the  second  member  is  under  the  radical  sign.  The  two 
values  of  the  unknown  quantity  Avill  be  alike,  except  that 
one  will  be  positive,  and  the  other  negative. 

316.  But  in  affected  quadratics,  a  part  only  of  one  side  of 
the  reduced  equation  is  under  the  radical  sign.  When  this 
part  is  added  to,  or  subtracted  from,  that  which  is  without 
the  radical  sign ;  the  two  results  will  differ  in  quantity,  and 
will  have  their  signs  in  some  cases  alike,  and  in  others  un- 
like. 

1 .  The  equation  ar'-f8a::=20 


Becomes  w^hen  reduced,         x=  -  4t\/l6-\-20. 

That  is  x=-4±e. 

Here  the  first  value  of  x  is,  -  4-f-6=:-f-2  >  one  positive,    and 
And  the  second  is  -4-6~-loJ  the  other  negative. 

19 


14G  ALGEBRA. 

2.  The  equation  a^^-Sx^  -  15 

Becomes  when  reduced,         x=4t\/\Q  -  15 
That  is  xz=4tl 

Here  the  first  value  of  x  is  4+1  =-\-5  )  ^^^-,   ^^gj^j^g 
And  the  second  is  4-1=4-3  3  P  *        • 

That  these  two  vahies  of  x  are  correctly  found,  may  be 
proved,  by  substituting  first  one  and  then  the  other,  for  x  it- 
self, in  the  original  equation.      (Art.  194.) 

Thus5'-8x5==25-40=-15 

And  3^-8x3  =  9-24= -15. 

317.  In  the  reduction  of  all  affected  quadratic  equation, 
the  value  of  the  unknown  quantity  is  frequently  found  to  be 
imaginary. 

Thus  the  equation  x"^  -Sx=  -  20 

Becomes,  when  reduced,  x=4t\^\6-  20 

That  is,  x=4±\/-4. 

Here  the  root  of  the  negative  quantity  -  4  can  not  be  as- 
signed, (Art.  263,)  and  therefore  the  value  of  x  can  not  be 
found.  There  will  be  the  same  impossibility,  in  every  in- 
stance in  which  the  negative  part  of  the  quantities  under  the 
radical  sign  is  greater  than  the  positive  part.* 

318.  Whenever  one  of  the  values  of  the  unknown  quan- 
tity, in  a  quadratic  equation,  is  imaginary,  the  other  is  so 
also.     For  both  are  equally  affected  by  the  imaginary  root. 

Thus  in  the  example  above 

The  first  value  of  x  is  4-|- V  -  4, 

And  the  second  is  4  -  \/  -  4  ;  each  of  which 
contains  the  imaginary  quantity  /\/  -  4. 

319.  An  equation  which  when  reduced  contains  an  ima- 
ginary root,  is  often  of  use,  to  enable  us  to  determine  whether 
a  proposed  question  admits  of  an  answer,  or  involves  an  ab- 
surdity. 

Suppose  it  is  required  to  divide  S  into  two  such  parts,  that 
the  product  will  be  20. 


♦  See  Note  G. 


QUADRATIC   EQUATIONS.  147 

If  X  is  one  of  the  parts,  the  other  will  be  8  -x.  (Art.  195.) 
By  the  conditions  proposed  (8  -  a:)  Xa;=20 

This  becomes,  when  reduced,  a;=4±\/-  4. 

Here  the  imaginary  expression  \/-4  shows  that  an  an- 
swer is  impossible  ;  and  that  there  is  an  absurdity  in  suppo- 
sing tliat  8  may  be  divided  into  two  such  parts,  that  their 
product  shall  be  20. 

320.  Although  a  quadratic  equation  has  two  solutions,  yet 
both  these  may  not  always  be  applicable  to  the  subject  pro- 
posed. The  quantity  under  the  radical  sign  may  be  produced 
either  from  a  positive  or  a  negative  root.  But  both  these  roots 
may  not,  in  every  instance,  belong  to  the  problem  to  be  sol- 
ved.    See  Art.  299. 

Divide  the  number  30  into  two  such  parts,  that  their  jm-o- 
duct  may  be  equal  to  8  times  their  difference. 

If  x=  the  lesser  part,  then  30-a;=  the  greater. 

By  the  supposition,  xx  (30  -  a:)  =8 X  (30  -  2x) 

This  reduced,  gives  a-=23±17=40  or  6=  the  lesser  part. 

But  as  40  cannot  be  a  part  of  30,  the  problem  can  have 
but  one  real  solution,  making  the  lesser  part  6,  and  the  greater 
part  24. 

Examples  of  Quadratic  Equations. 

1.  Reduce  3x'-9j;- 4=80.  Ans.  x=7,  or -4. 

2.  Reduce  4x  -  ?illf  =  46.  Ans.  a;=12,  or-  |. 

X 

1  4.  —   -7' 

3.  Reduce  4x- =14.  Ans.  x=4,  or  -i. 

a:-|-l 

4.  Reduce   5a:-^^~^  =  2a:-|-^^-l^.       Ans.  x=4,  or  -  1. 

a:-3  2 

5.  Reduce   l^-l^-?^  =  3.  Ans.  a; =4,  or  2J^. 

X  4x^ 

6.  Reduce  ^^"^1  =  10-  —  .  Ans.  a;=12,  or  6. 

x-4  2 

7.  Reduce  =^-.1^1=^^1-1.      Ans.  «=21,  or  5. 

3       x-S         9 


148  ALGEBRA. 

8.  Reduce  '—l^^L±l  =  x-S,  Ans.  x=h  or -28 

x'-  6x-f9 

9.  Reduce  _?-+?=S.  Ans.  x=2, 

x-\-l      X 

10.  Reduce  1^-^!LlL=x-9.  Ans.  ar^lO. 

x+2        6 

11.  Reduce -+-=?.  Ans.  x=\t\/r^'' 

a     x     a 


12.  Reduce  x'-i-ax'=b.   Ans.  xz=  (  -|-\/^+lY 

13.  Reduce  ^"-f!=-i-.  Ans.  a:=Vi- 

2      4  32 

14.  Reduce  2x^+3x^=2.  Ans.  x=i. 

15.  Reduce  ix-i^x=22i.  Ans.  a:=49. 

16.  Reduce2a?*-a;''+96  =  99.  Ans.  a:=iV6.     ' 

17.  Reduce  (10+a:)2-(10+a;)^=:2.    Ans.  x=6. 

18.  Reduce  3a;'"-2a;"=8.  Ans.  a:=^2. 

19.  Reduce  2(l+ar-ar^)  -\/l+x-x'=  -i. 

Ans.  a:=i+iV4^. 

20.  Reduce  V^^-a3=a;- 6.  A"^- ^=2-V  "Ig^"* 

21.  Reduce  Vf^^lz^l  Ans.  x=4, 

22.  Reduce  x^+x^=z75G.  Ans.  a:=243. 

21 


23.  Reduce  aJ2x+\+2\/x=  _  -^ .     Ans.  x=4. 

^  V2a:+1 

24.  Reduce  2a^'^^+S\/2x=-±^.     Ans.  a:=9a. 


25.  Reduce  x+16-7Va:+16  =  10-4Va:4-16.  Ans.a:=9 

26.  Reduce  V^+V^= 6 V^- 

Dividing  by  Vx,         a^-\-x  =  6.  Ans.  x=2. 


QUADRATIC    EQUATIO:^ 

nS.                     149 

27. 

Reduce 

4a-- 5     3.r-T_9.r-l-23 
X         3x4-7        13a; 

Ans.  x=2. 

08 

Reduce 

3      _L      6      _11 

Ans.  rr  — 3. 

Qx-x'   '    x^-\-2x     ox 

29. 

Reduce 

{x-5)  -S{x-5y  =40. 

Ans.  a- =9. 

30.  Reduce    rr-|-V^-f^  =  ^+2  V^+6-       -A.ns.  rc^lO. 
PROBLEMS  PRODUCING  aUADRATIC  EQUATIONS. 

Prob.  1.  A  merchant  has  a  piece  of  cotton  clotli,  and  a 
piece  of  silk.  The  number  of  yards  in  both  is  110  :  and  if 
the  square  of  the  number  of  yards  of  silk  be  subtracted  from  80 
times  the  number  of  yards  of  cotton,  the  difference  will  be 
400.     How  many  yards  are  there  in  each  piece  1 

Let  x=2z  the  yards  of  silk. 
Then  110-rr=:  the  yards  of  cotton. 
By  supposition       400  z=  80  x  ( H  0  -  .t)  -  x- 
Therefore  x=  -  40±Vioooo=  -  40±100. 

The  first  value  of  x,  is  -  40+100  =  60,  the  yards  of  silk; 
And  110 -a:=110- 60  =  50,  the  yards  of  cotton. 

The  second  value  of  .t,  is  -  40  -  100=  -  140  ;  but  as  this 
is  a  negative  quantity,  it  is  not  applicable  to  goods  which  a 
man  has  in  his  possession. 

Prob.  2.  The  ages  of  two  brothers  are  such,  that  their  sum 
is  45  years,  and  their  product  500.   What  is  the  age  of  each  1 

Ans.  25  and  20  years. 

Prob.  3.  To  find  two  numbers  such,  that  their  difference 
shall  be  4,  and  their  product  117. 

Let  a;=  one  number,  and  x-{-4=  the  other. 

By  the  conditions  (^^+4)  X''^=  1 17. 

This  reduced,  gives  a:  =  -  2±\/l2l  =  -  2±1 1 . 

One  of  the  numbers  therefore  is  9,  and  the  other  13. 

Prob.  4.  A  merchant  having  sold  a  piece  of  cloth  which 
cost  him  30  dollars,  found  that  if  the  price  for  which  he  sold 
it  were  multiplied  by  his  gaAn,  the  product  would  be  equal  to 
the  cube  of  his  gain.     What  was  his  gain  1 


150  ALGEBRA. 

Let  x=  the  gain. 

Then  30-[-3-=  the  price  for  which  the  cloth  was  sold. 
By  the  statement  r'^  (30+ a:)  x^ 

Therefore  x  =  i±Vi+30  =  |  ±V- 

The  first  vahie  of  x  is  i-r\^  =-[-6.  > 

The  second  value  is  ^  -  \^  =  -  5.  ) 

As  the  last  ans\ver  is  negative,  it  is  to  he  rejected  as  incon- 
sistent with  the  nature  of  the  problem,  (Art.  320.)  for  gain 
must  be  considered  positive. 

Prob.  5.  To  find  two  numbers  whose  difference  shall  be  3, 
and  the  difference  of  their  cubes  117. 

Let  x=  the  less  number. 
Then  x-\-S  =  the  greater. 

By  supposition  (.T4-3)^-a;^=117 

Expanding  {x-{-Sy  (Art.  217.)  9a-^+27a;  =  n7- 27=90 

And  x=- 1- ±  V  V-  =  -2±i' 

The  t^vo  numbers,  therefore,  are  2  and  5. 

Prob.  6.  To  find  two  numbers  -whose  difference  shall  be 
12,  and  the  sum  of  their  squares  1424. 

Ans.  The  numbers  are  20  and  32. 

Prob.  7,  Two  persons  draw  prizes  in  a  lottery,  the  differ- 
ence of  which  is  120  dollars,  and  the  greater  is  to  the  less, 
as  the  less  to  10.     What  are  the  prizes  ? 

Ans.  40  and  160. 

Prob.  8.  What  two  numbers  are  those  whose  sum  is  6,  and 
the  sum  of  their  cubes  72  1  Ans.  2  and  4. 

Prob.  9.  Divide  the  number  56  into  two  such  parts,  that 
their  product  shall  be  640. 

Putting  X  for  one  of  the  parts,  we  have,  a;  =  28j:12  =  40  or 
16. 

In  this  case,  the  two  values  of  the  unknown  quantity  are 
the  tw^o  parts  into  w^iich  the  given  number  w^as  required  to 
be  divided. 

Prob.  10.  A  gentleman  bought  a  number  of  pieces  of  cloth 
for  675  dollars,  which  he  sold  again  at  48  dollars  by  the  piece, 
and  gained  by  the  bargain  as  much  as  one  piece  cost  him. 
What  was  the  mmiber  of  pieces'?  Ans.  15. 


QUADRATIC  EQUATIONS.  151 

Prob.  II.  ..3  and  B  started  together,  for  a  place  150  miles 
distant.  A''s  hourly  progress  was  3  miles  more  than  5's,  and 
he  arrived  at  his  journey's  end  8  hours  and  20  minutes  before 
B.     What  was  the  hourly  progress  of  each  ? 

Ans.  9  and  6  miles. 

Prob.  12.  The  difference  of  two  numbers  is  6  ;  and  if  47 
be  added  to  twice  the  square  of  the  less,  it  will  be  equal  to 
the  square  of  tlie  greater.     What  are  the  numbers  % 

An?.  Hand  11. 

Prob.  13.  A  and  B  distributed  1200  dollars  each,  among 
a  certain  number  of  persons.  A  rehcved  40  persons  more 
than  B,  and  B  gave  to  each  individual  5  dollars  more  tlian 
A.     How  many  were  relieved  by  A  and  B  ? 

Ans.  120  by  A,  and  80  by  B. 

Prob.  14.  Find  two  niunbers  whose  sum  is  10,  and  the  sum 
of  their  squares  6S.  Ans.  7  and  3. 

Prob.  15.  Several  gentlemen  made  a  pin-chase  in  company 
for  175  dollars.  Tw^o  of  them  having  wit hdraw^n,  the  bill 
was  paid  by  the  others,  each  fmnishing  10  dollars  more  than 
would  have  been  his  equal  share  if  the  bill  had  been  paid  by 
the  -whole  company.  What  was  the  number  in  the  company 
at  first  1  Ans.  7. 

Prob.  16.  A  merchant  bought  severa,l  yards  of  linen  for 
60  dollars,  out  of  which  he  reserved  15  yards,  and  sold  the 
remainder  for  54  dollars,  gaining  10  cents  a  yard.  How 
many  yards  did  he  buy,  and  at  what  price  ] 

Ans.  75  yards,  at  80  cents  a  3^ard. 

Prob.  \1,  A  and  B  set  out  from  two  towns,  Avhich  were 
247  miles  distant,  and  travelled  the  direct  road  till  they  met. 
A  went  9  miles  a  day  ;  and  the  num.bcr  of  days  which  they 
travelled  before  meeting,  was  greater  by  3,  than  the  number 
of  miles  which  B  went  in  a  day.  How  many  miles  did  each 
travel  1  Ans.  A  went  117,  and  B  130  miles. 

Prob.  18.  A  gentleman  bought  two  pieces  of  cloth,  the 
finer  of  w^iich  cost  4  shillings  a  yard  more  than  the  other. 
The  finer  piece  cost  £18 ;  but  the  coarser  one,  which  was  2 
yards  longer  than  the  finer,  cost  only  £16.  How  many 
yards  were  there  in  each  piece,  and  what  was  the  price  of  a 
yard  of  each  1 

Ans.  There  were  18  yards  of  the  finer  piece,  and  20  of  the 
coarser  ;  and  the  prices  were  20  and  16  shillings. 


152  ALGEBRA. 

Prob.  19.  A  merchant  bought  54  gallons  of  Madeira  wine, 
and  a  certain  quantity  of  Tenerifle.  For  the  former,  Re  gave 
half  as  many  shillings  by  the  gallon,  as  there  were  gallons 
of  Tenerifie,  and  for  the  latter,  4  shilhngs  less  by  the  gallon. 
He  sold  tlie  mixture  at  10  shillings  by  the  gallon,  and  lost 
428  16s.  by  his  bargain.  Required  the  price  of  the  Madeira, 
and  the  number  of  gallons  of  Tenerifle. 

Ans.  The  Madeira  cost  18  shillings  a  gallon,  and  there 
were  36  gallons  of  Teneriffe. 

Prob.  20.  If  the  square  of  a  certain  number  be  taken  from 
40,  and  the  square  root  of  this  difference  be  increased  by  10, 
.ind  the  sum  be  multiplied  by  2,  and  the  product  divided  by 
the  number  itself,  the  quotient  will  be  4.  What  is  the 
number  1  Ans.  6. 

Prob.  21.  A  person  being  asked  his  age,  replied,  If  you 
add  the  square  root  of  it  to  half  of  if,  and  subtract  12,  the 
remainder  will  be  nothing.    What  was  his  age  ? 

Ans.  16  years. 

Prob.  22.  Two  casks  of  wine  were  purchased  for  58  dol* 
lars,  one  of  which  contained  5  gallons  more  than  the  other, 
and  the  price  by  the  gallon,  was  2  dollars  less  than  ^  of  the 
number  of  gallons  in  the  smaller  cask.  Required  the  num- 
ber of  gallons  in  each,  and  the  price  by  the  gallon. 

Ans.  The  numbers  were  IS  and  17,  and  the  price  by  the 
gallon  2  dollars. 

Prob.  23.  In  a  parcel  which  contains  24  coins  of  silver  and 
copper,  each  silver  coin  is  worth  as  many  cents  as  there  are 
copper  coins,  and  each  copper  coin  is  worth  as  man}^  cents  as 
there  are  silver  coins  ;  and  the  whole  are  worth  2  dollars  and 
1 6  cents.     How  many  are  there  of  each  1 

Ans.  6  of  one,  and  18  of  the  other. 

Prob.  24.  A  person  bought  a  certain  number  of  oxen  for 
80  guineas.  If  he  had  received  4  more  oxen  for  the  same 
money,  he  would  have  paid  one  guinea  less  for  each.  What 
was  the  number  of  oxen  ?  Ans.   1 6. 

SUBSTITUTION. 

321.  In  tne  reduction  of  Quadratic  Equations,  as  well  as 
in  other  parts  of  Algebra,  a  complicated  process  may  be  ren- 
dered much  more  simple,  by  introducing  a  new  letter  which 


QUADRATIC  EQUATIONS.  153 

shall  be  made  to  represent  several  others.  This  is  termed 
substitutiori.  A  letter  may  be  put  for  a  compound  quantity 
as  well  as  for  a  single  number.     Thus  in  the  equation 

we  may  substitute  6,  for  f-[- V^^  -  64-\-h.  The  equation 
will  then  become  x--2ax=b,     and  when  reduced 

will  be  x=a±\/d--\-b. 

After  the  operation  is  completed,  the  compound  quantity 
for  which  a  single  letter  has  been  substituted,  may  be  restor- 
ed. The  last  equation,  by  restoring  the  value  of  b,  will  be- 
come 


Reduce  the  equation     ax  -  2x  —  dz=bx~  x~  -  x 
Transposing,  &c.  x^-]-(a-  b  -l)x^  =  d 

Substituting /i  for  (a-b-l),  a^J^hx=d 


h4-      /fi^ 
Therefore  x=  -^_  v/  ^+f^ 


Restoring  the  value  of  /i,  x=  _a-b-l -{-      /(a-b-]) 


SECTION  XL 


SOLUTION  OF  PROBLEMS  WHICH  CONTAIN  TWO 
OR  MORE  UNKxNOWN  QUANTITIES. 

DEMONSTRATION  OF  THEOREMS. 

Art.  322.  IN  the  examples  which  have  been  given  of  the 
resolution  of  equations,  in  the  preceding  sections,  each  pro- 
blem has  contained  only  one  unknown  quantity.  G<  if,  in 
some  instances,  there  have  been  two,  they  have  beeb  no  re- 
lated to  each  other,  that  they  have  both  been  expre.  ed  by 
means  of  the  same  letter.     (Art.  195.) 

20 


154  ALGEBRA. 

But  cases  frequently  occur,  in  which  several  unknown 
quantities  are  introduced  into  the  same  calculation.  And  if 
the  problem  is  of  such  a  nature  as  to  admit  of  a  determinate 
answer,  there  will  arise  from  the  conditions,  as  many  equa- 
tions independent  of  each  other,  as  there  are  unknown  quan- 
tities. 

Equations  are  said  to  be  independent,  when  they  express 
different  conditions ;  and  dependent,  when  they  express  the 
same  conditions  under  different  forms.  The  former  are  not 
convertible  into  each  other.  But  the  latter  maybe  changed 
from  one  form  to  the  other,  by  the  methods  of  reduction 
which  have  been  considered.  Thus  b  -  x=y,  and  b=zy-\-x, 
are  dependent  equations,  because  one  is  formed  from  the 
other  by  merely  transposing  x. 

323.  In  solving  a  problem,  it  it  necessary  first  to  find  the 
value  of  one  of  the  unknown  quantities,  and  then  of  the 
others  in  succession.  To  do  this,  we  must  derive  from  the 
equations  which  are  given,  a  new  equation,  from  which  all 
the  unknown  quantities  except  one  shall  be  excluded. 

Suppose  the  following  equations  are  given. 

1.  x-\-y=zl4: 

2.  x-y  =  2. 

If  y  be  transposed  in  each,  they  will  become 

1.  x=\4-y  • 

2.  x=2-{-y. 

Here  the  first  member  of  each  of  the  equations  is  x,  and 
the  second  member  of  each  is  equal  to  x.     But  according  to 
axiom  11th,  quantities  which  are  ref^-^f^ctively  equal  to  any 
other  quantity  are  equal  to  each  other  ;  therefore, 
2-]-y=U-y, 

Here  we  ha;i^e  a  new  equation,  which  contains  only  the 
unknown  quantity  y.     Hence,        *' 

324.  Rule  I.  To  exterminate  one  jf  tno  unknown  quan- 
tities, and  deduce  one  equation  frofit  two  ;  Find  t  je  value 

OF  ONE  OF  THE  UNKNOWN  QUANTITIES  IN  EACH  O^^  THE  EQUA- 
TIONS. AND  FORM  A  NEW  EQUATION  BY  MAKING  ONE  OF  THESE 
VALUES  EQUAL  TO  THE  OTHER. 

That  quantity  which  is  the  least  involved  should  be  the 
one  wliich  is  chosen  to  be  exterminated. 


EQUATIONS.  155 

For  the  convenience  of  referring  to  different  parts  of  a  so- 
lution, the  several  steps  will,  in  future  be  numbered.  When 
an  equation  is  formed  from  one  immediately  preceding,  it  wilt 
be  unnecessary  to  specify  it.  In  other  cases,  the  number  of 
the  equation  or  equations  from  which  a  new  one  is  derived, 
will  be  referred  to. 

Prob.  1.  To  find  two  numbers  such,  that 
Their  sum  shall  be  24 ;  and 
The  greater  shall  be  equal  to  five  times  the  les<i. 
Let  a;=the  greater;  And  y  =  ihe  less. 

1.  By  the  first  condition,  x-\-y  =  24} 

2.  By  the  second,  x=5y        ) 

3.  Transp.  y  in  the  first  equation,  x  =  2i-y 

4.  Making  the  2d  and  3d  equal,  5i/  =  24-j/ 

5.  And  2/ =  4,  the  less  number. 

Prob.  2.   To  find  one  df  two  quantities, 
Whose  sum  is  equal  to  h ;  and 
The  difference  of  whose  squares  is  equal  to  d. 

Let  x=  the  greafer  quantity  ;  And  y=  the  less. 

1.  By  the  first  condition,  x-]-y  =  h      > 

2.  By  the  second,  x^-y'^=d    ] 

3.  Transp.  y'^  in  the  2d  equation,  x^^d-^-y"^ 


4.  By  evolution,  (Art.  297.)  x=^d-\-y'' 

5.  Trans,  y  in  the  first  equation,  x  =  h-y 

6.  Making  the  4th  and  5th  equal,  ^d-{-y'^=h-  y 

7.  Therefore  y     ^'"^ 


2h 

Twh.S.  Given  axJrby=h    ?  To  find  y.     An^.  y^^^l^. 
And      x-\-y  =  a     5  ^  ^      b^a 

325.  The  rule  given  above  may  be  generally  applied,  for 
the  extermination  of  unknown  quantities.  But  there  are 
cases  in  which  other  methods  v/ill  be  found  more  expeditious. 

Suppose  x=hy       ) 
And  ax-\-bx=y^  y 

i 


156  ALGEBRA. 

As  in  the  first  of  these  equations  x  is  equal  to  hy,  we  may 
in  the  second  equation  substitute  tliis  vahie  of  x  instead  of 
X  itself.  The  second  equation  will  then  be  converted  into 
ahy-irbhy  =  y\ 

The  equality  of  the  two  sides  is  not  affected  by  this  alter- 
ation, because  we  only  change  one  quantity  x  for  another 
which  is  equal  to  it.  By  this  means  we  obtain  an  equation 
which  contains  only  one  unknown  quantity.     Hence, 

326.  Rule  II.  To  exterminate  an  unknown  quantity,  find 

THE  VALUE  OF  ONE  OF  THE  UNKNOWN  QUANTITIES,  IN  ONE  OF 

THE  EQUATIONS  ;  and  then  in  the  other  equation  SUBSTI- 
TUTE THIS  VALUE  FOR  THE  UNKNOWN  QUANTITY  ITSELF. 

Problem  4.  A  privateer  in  chase  of  a  ship  20  miles  distant, 
sails  8  miles,  while  the  ship  sails  7.  How  far  must  the  pri- 
vateer sail  before  she  overtakes  the  ship  1 

It  is  evident  that  the  whole  distance  which  the  privateer 
sails  during  the  chase,  must  be  to  the  distance  which  the 
ship  sails  in  the  same  time,  as  8  to  7. 

Let  x=  the  distance  which  the  privateer  sails  : 
And  y=  the  distance  w^hich  the  ship  sails. 

1.  By  the  supposition,  x=y-\-20  ) 

2.  And  also,  x  :y  :  :8:7  ) 

3.  Art.  188,  y=ix 

4.  Substituting  t  for  y,  in  the  1st  equation,  x=ix-\-20 

5.  Therefore,  a;=160. 

Prob.  5.  The  ages  of  two  persons,  A  and  B,  are  such  that 
seven  years  ago,  A  was  three  times  as  old  as  B;  and  seven 
years  hence,  A  will  be  twice  as  old  as  B.  What  is  the  age 
of  B? 

Let  x=  the  age  of  A;         And  y  =  ihe  age  of  B. 
Then  x-7  was  the  age  of  A,  7  years  ago  ; 
And  y  -7  was  the  age  of  B,  7  years  ago  ; 
Also  x-\-7  will  be  the  age  of  A,  7  years  hence  ; 
And  y-{-7  will  be  the  age  of  B,  7  years  hence. 

1.  By  the  first  condition,  x -7  =  Sx(y -7)=3y -2\  ( 

2.  By  the  second,  x-{-7=2x{y-\-7)=2y-{-14  ] 

3.  Transp.  7  in  the  1st  equa.     x=Sy  -  14 

4.  Subst.  3i/-  14  for  x,  in  the  2d,  3i/ -  14-f  7=2i/4-14 

5.  Therefore,  t/  =  21,  the  age  of  B. 


EQUATIONS  157 

Prob.   6.   There  are  two  number?,  of  wliich, 

The  greater  is  to  the  less  as  3  to  2  ;  and 
Their  sum  is  tlie  6th  part  of  their  product. 

What  is  the  less  number  1  Aus.  10. 

327.  There  is  a  third  method  of  exterminal  iug-  an  unkaown 
quantity  from  an  equation,  which  in  many  cases,  is  preferable 
to  either  of  the  preceding. 

Suppose  that  x-\-Sy  =  a  > 
And  x-Sy  =  b  ) 

If  we  add  together  the  first  members  of  these  two  equa- 
tions, and  also  the  second  members,  we  shall  have 

2x=a-\-b, 

an  equation  which  contain?  only  the  unknown  quantity  x. 
The  other,  having  equal  co-efficients  with  contrary  signs,  has 
disappeared.  (Art.  77.)  The  equality  of  the  sides  is  preserved 
because  we  have  only  added  equal  quantities  to  equal  quan- 
tities. 

Again,  suppose  ox-\-y  =  h 

And  2x-\-y—d 

If  we  subtract  the  last  equation  from  the  first,  we  shall  have 

x=h-d 

where  y  is  exterminated,  without  affecting  the  equality  of 
the  sides. 

Again,  suppose  x  —  2y  =  a  ) 

And  x-\-4y  —  b  > 

Multiplying  the  1st  by  2,     2x  -4y  =  2a 

Then  adding  the  2d  and  3d,  Sx=zb-]-2a.     Hence, 

328.  Rule  III.  To  exterminate  an  unknown  cpiantity, 
MULTIPLY  OR  DIVIDE  the  equations,  if  necessary, 

IN  SUCH  A  MANNER  THAT  THE  TERM  WHICH  CONTAINS  ONE 
OF  THE  UNKNOWN  QUANTITIES   SHALL  BE  THE   SAME  IN  BOTH. 

Then  SUBTRACT   one  equation  from  the  other, 

IF    the    SIGNS    OF    THIS    UNKNOWN    QUANTITY    ARE    ALIKE, 

or  add  them  together,  if  the  signs  are  UNLIKE. 

It  must  be  kept  in  mind  that  both  members  of  an  equa- 
tion are  always  to  be  increased  or  diminished,  multiplied  or 
divided  alike.*  (Art.  170.) 


158  ALGEBRA. 

Prob.  7.  The  numbers  in  two  opposing  armies  are  such, 
that, 

The  sum  of  both  is  21 1 10  ;  and 

Twice  the  number  in  the  greater  army,  added  to  tliree 
times  the  number  in  the  less,  is  52219. 

What  is  tlie  number  in  the  greater  army  ? 

Let  x=  the  greater.         And  y=  the  less. 

1.  By  the  first  condition,  x-\-y  =  2U\0      > 

2.  By  the  second,  2a:-f-3?/r^52219  S 

3.  Multiplying  the  1st  by  3,        Sx-\-Sy=6SSS0 

4.  Subtracting  the  2d  from  the  3d,  a;=  1 1 1 1 1. 

Prob.  8.  Given  2x-\-y=\6,  and  Sx-Sy-6,  to  find  the 
value  of  X. 

1.  By  supposition,  2x-]-y=16  } 

2.  And  Sx-Sy  =  6l 

3.  Multiplying  the  1st  by  3,  6x^Sy  =  48 

4.  Adding  the  2d  and  3d,  9a; =54 

5.  Dividing  by  9,  x=6. 

Prob.  9.  Given  x-\-y=14,  and  x-y  =  2,  to  find  the  value 
of  y.  Ans.  6. 

In  the  succeeding  problems,  either  of  the  three  rules 
for  exterminating  unknown  quantities  will  be  made  use  of,  as 
will  in  each  case  be  most  convenient 

329.  When  one  of  the  unknown  quantities  is  determined,  the 
other  may  be  easily  obtained,  by  going  back  to  an  equation 
which  contains  both,  and  substituting  instead  of  that  which 
is  already  found,  its  numerical  value. 

Prob.  10.  The  mast  of  a  ship  consists  of  two  parts  : 

One  third  of  the  low^er  part  added  to  one  sixth  of  the 
upper  part,  is  equal  to  28  ;  and. 

Five  times  the  lower  part,  diminished  by  six  times  the 
upper  part,  is  equal  to  12. 

What  is  the  height  of  the  mast  1 


EQUATIONS.  159 

Let  x=  the  lower  part ;         And  y=  the  upper  part. 

1.  By  the  first  condition,  hx-\-lij  =  2S  ) 

2.  By  the  second,  5x-6y  =  \2) 

3.  Multiplying  the  1st  by  6,  2x+y  =  lQS 

4.  Dividing  the  2d  by  6,  _        Ix-y=z2 

5.  Adding  the  3d  and  4tii,  2.r-f|a:  =  170 

6.  Multiplying  by  6,  12a:+5a:=1020 

7.  Uniting  terms  and  dividing  by  17,  x=60,  the  lower  part. 

Then  by  the  3d  step,  2x-{-y  =  \6S 

That  is,  substituting  60  for  x,  120+?/=  168         [per  part. 

Transposing  120,  i/  =  1 68  -  120=48,  the  up- 

Prob.  11.  To  find  a  fraction  such  that. 

If  a  unit  be  added  to  the  numerator,  the  fraction  will  be 
equal  to  {  ;  but 

If  a  unit  be  added  to  the  denominator,  the  fraction  will  be 
equal  to  i. 

I^et  x=  the  numerator,         Andt/=  the  denominator. 


1.  By  the  first  condition, 


y 


By  the  second.  _ ^  =  i 

3.  Therefore  x=4,  the  numerator. 

4.  And  ?/=  15,  the  denominator. 
Prob.  12.  What  two  numbers  are  those. 

Whose  difference  is  to  their  sum,  as  2  to  3 ;  and 
Whose  sum  is  to  their  product,  as  3  to  5  *? 

Ans.  10  and  2. 

Prob.  13.  To  find  two  numbers  such,  that 

The  product  of  their  sum  and  difference  shall  be  5,  and 
The  product  of  the  sum  of  their  scjuares  and  the  differ- 
ence of  their  squares  shall  be  65. 

Let  x=  the  greater  number  ;         Andy=  the  less. 


160  ALGEBRA. 

1.  By  the  first  condition,  {^-\-y)x{x-y)  =  5        ) 

2.  By  the  second,  l^'-hy)  X  {x'  -  vl  =  65  ^ 

3.  Mult,  the  factors  in  the  1st,  (Art.  235,)  x^  -y''=5 

4.  Dividing  the  2d  by  the  3d,  (Art.  1 18,)  rr'+7/'=:13 

5.  Adding  the  3d  and  4th,  2a;2=  18 

6.  Therefore  x=S,  the  greater  number, 

7.  And  y  =  ^y  lliG  less. 

In  the  4th  step,  tlie  first  member  of  the  second  equation  is 
divided  by  x^  -  y^,  and  the  second  member  by  5,  whicli  is 
equal  to  x^  -  if. 

Prob.  14.  To  find  two  numbers  whose  differejice  is  8,  and 
product  240. 

Prob.  15.  To  find  two  numbers. 

Whose  difference  shall  be  12,  and 
The  sum  of  their  squares  1424. 

Let  x=  the  greater  ;  And  y=  the  less. 

1 .  By  the  first  condition,  x-y=l2  > 

2.  By  the  second,  x''-\-if=U24    5 

3.  Transposing  y  in  the  first,        x=y^l2  I 

4.  Squaring  both  si.des,  a;^=?/^-j-242/-f-144 

5.  Transposing  ?/^  in  the  second,  x'^=l42i-y^ 

6.  Making  the  4th  and  5thequal,?/^-|-24t/4- 144=  1424  -  y^ 

7.  Therefore  y=  -  6t^{67G)  =  -  6±26 

8.  And  a'=7/4-12  =  20+12  =  32. 

EaUATIONS    WHICH    CONTAIN   THREE   OR  MORE 
UNKNOWN   aUANTITIES. 

330.  In  the  examples  hitherto  given,  each  has  contamed 
no  more  than  tico  unknown  quantities.  And  two  indepen- 
dent equations  have  been  sufficient  to  express  the  conditions 
of  the  question.  But  problems  may  involve  three  or  more 
unknown  quantities ;  and  may  require  for  their  solution  as 
many  independent  equations. 

Suppose  .T-f-i/-]- 2^=1 2      ^ 

And        ^-\-2y  -  2^=  10  \  are  given  to  find,  x,  r/,  and  r* 

And        x-\-y-z  =  4:         ) 


EQUATIONS.  161 

From  these  three  equations,  two  others  may  be  derived 
which  shall  contain  only  two  unknown  quantities.  One  of 
the  three  in  the  orig-inal  equations  may  be  exterminated,  in 
the  same  manner  as  when  there  are,  at  first,  only  two,  by  the 
rules  in  Arts.  324,  6,  8. 

In  the  equations  given  above,  if  we  transpose  y  and  r,  w© 
shall  have. 

In  the  first,  x=\2-y-  z  ^ 
In  the  second,  a:=r:  10 -  2y-\~2z  \ 
In  the  third,      x=4-y-{-2         ) 

From  these  we  may  deduce  two  new  equations,  from  which 
X  shall  be  excluded. 

By  making  the  1  st  and  2d  equal ,    \2-y-z=\0- 2y-f- 2z  > 
By  making  the  2d  and  3d  equal,     lO-2y-^2z  =  4-y^z    > 
Reducing  the  first  of  these  two,      y  =  Sz-2  > 
Reducing  the  second,  y=z-]-6    ) 

From  these  two  equations  one  may  be  derived  containing 
only  one  unknow:  i  quantity 

Making  one  equal  to  the  other,         Sz'-.2=z-\-6 
And  .  z=4.     Hence, 

331.  To  solve  a  problem  containmg  three  imknown  quan- 
tities, and  producing  three  independent  equations. 

First,  from  the  three  equations  deduce  two  con- 
taining ONLY   two    unknown    QUANTITIES. 

Then,  from  these  two  deduce  one,  containing  onlt 
one  unknown  quantity. 

For  making  these  reductions,  the  rules  already  given  are 
sufficient.     (Art.  324,  6,  8.) 

Prob.  16.  Let  there  be  given, 

1.  The  equation  x-\-5y-{-6z=DS  J 

2.  And  a:+3i/4-3z  =  30  V  To  find  x,  y,  and  z. 

3.  And  x-lry-\-z=l2      ) 

From  these  three  equations  to  derive  two,  containing  only 
two  unknown  quantities, 

4.  Subtract  the  2d  from  the  Ist,    2i/-|-3z=23) 

5.  Subtract  the  3d  from  the  2d,     2y-\-2z:=i  18  \ 
From  these  two,  to  derive  one, 

6.   Subtract  the  5ih  from  the  4th,  z=5. 

21 


162  ALGEBRA. 

To  find  X  and  t/,  we  have  only  to  take  their  vahies  from 
the  third  and  fifth  equations.    (Art.  329.) 

7.  Reducing  the  fifth,         1/^=9-^=9-5  =  4 

8.  Transposing  in  the  third,  x—\2  - z  -y  =  \2  -  5  -  4t  =  ^. 

Prob.  17.  To  find  rr,  y,  and  r,  from 

1.  The  equation  x-\-y-\-z=\'i      ^ 

2.  And  x^2y-\-^z  =  %0\ 

3.  And  ia:+ii/+2=6    ) 

4.  Multiplying  the  1st  by  3,  3.r+3?/+32=36 

5.  Subtracting  the  2d  from  the  4th,    2x-\-y=\Q 

6.  Subtracting  the  3d  from  the  1st,    x-^x-\-y-^=Q 

7.  Clearing  the  6th  of  fractions,         4a;4-3y  =  36  > 

8.  Multiplying  the  5th  by  3,  ^x-\-^  =  AS  ] 

9.  Subtracting  the  7th  from  the  8th,  2a:  =12.     And  a: =6. 

ir.    r»  J     •       .X     .7.1  36 -4a;     36-24      . 

10.  Reducmg  the  7th,  ?/  = — = — - — =4. 

o  o 

11.  Reducing  the  1st,  ^=12-a;-j/=12- 6 -4=2. 

In  this  example  all  the  reductions  have  been  made  accor- 
ding to  the  third  rule  for  exterminating  imknown  quantities. — 
(Art  328. )  But  either  of  the  three  may  be  used  at  pleasure. 

332.  A  calculation  may  often  be  very  much  abridged,  by 
the  exercise  of  judgment  in  stating  the  question,  in  selecting 
the  equations  from  which  others  are  to  be  deduced,  in  simpli- 
fying fractional  expressions,  in  avoiding  radical  quantities, 
&c.  The  skill  which  is  necessary  for  this  purpose,  hoAvever, 
is  to  be  acquired,  not  from  a  system  of  rules,  but  from  prac- 
tice, and  a  habit  of  attention  to  the  peculiarities  in  the  con- 
ditions of  different  problems,  the  variety  of  ways  in  which 
the  same  quantity  may  be  expressed,  the  numerous  forms 
which  equations  may  assume,  &c.  In  many  of  the  examples 
in  this  and  the  preceding  sections,  the  processes  might  have 
been  shortened.  But  the  object  has  been  to  illustrate  gen- 
eral principles  rather  than  to  furnii^li  specimens  of  expeditious  ^ 
solutions.  The  learner  will  do  well,  as  he  passes  along,  to 
exercise  his  skill  in  abridging  the  calculations  which  are 
here  giveu,  or  substituting  others  in  their  stead. 


EQUATIONS.  1G3 

C  1.  x-\-y  =  a  ^ 
Prob.  18.  Given    <  2.  x-\-z  =  b>    To  find  x,  y  and  z. 

^3.  y-{-z=c) 
a-\-b-c                   a-\-c-b  fe-j-c-ct 

Ans.  x= 2 -^^^^  y  = {> ^^^"  -—       2 — ' 

Prob.  19.  Three  persons  .5,  B,  and  C,  purchase  a  horse 
for  100  dollars,  but  neither  is  able  to  pay  for  the  whole. 
The  payment  would  require, 

The  whole  of  ^Ts  money,  together  with  half  of  J5's  ;  or 

The  whole  of  ^'s,  with  one  third  of  (7s  ;  or 

The  whole  of  Cs,  with  one  fourth  of  A's. 
How  much  money  had  each  1 

Let  x=A^s 
y  =  B^s 
By  the  first  condition, 
By  the  second. 
By  the  third, 
Therefore        a: =64.       y-. 

333.  The  learner  must  exercise  his  own  judgment,  as  to 
the'  choice  of  the  quantity  to  be  first  exterminated.  It  will 
generally  be  best  to  begin  with  that  which  is  most  free  from 
co-efficients,  fractions,  radical  signs,  &c. 

Prob.  20.  Tlie  sum  of  the  distances  which  three  persons, 
./?,  B,  and  C,  have  travelled,  is  62  miles ; 
.^'s  distance  is  equal  to  4  times  Cs,  added  to  twice  ^'s  ;  and 
Twice  .^'s  added  to  3  times  ^'s,  is  equal  to  17  times  C's. 

What  are  the  respective  distances  ? 

Ans.  c^'s,  46  miles  ;  B%  9  ;   Cs  7. 

Prob.  21.  To  find  a-,  y,  and  r,  from 

The  equation  ix-\-iy-^iz=62  ^ 

And  ix^iyJriz=47  \ 

And  4a:+ii/+iz  =  38) 

Ans.  a:=:24  t/=:60.  z=\20. 

Cxy=600^ 
Prob.  22.   Given   <  a:^— 300  >    To  find  x,  y,  and  z, 

(yzz=200) 
Ans.  a;  =  30.  y  =  20.  z=10. 


164  ALGEBRA. 

334.  The  same  method  which  is  employed  for  the  reduc- 
tion of  three  equations,  may  be  extended  to  4,  5,  or  any  num- 
ber of  equations,  containing  as  many  unknown  quantities. 

The  unknown  quantities  may  be  exterminated,  one  after 
another,  and  the  number  of  equations  may  be  reduced  by 
successive  steps  from  five  to  four,  from  four  to  three,  from 
three  to  two,  &c. 

Prob.  23.  To  find  tr,  x,  y,  and  z,  from 
1.  The  equation  ■fty-\.z^iiD  =  S  \ 

!•  -^"'}  "iy-t^'^t     }  Four  ;quation8. 

3.  And  x-\-y-{-2=\2     C  * 

4.  And  x-\-w-{-z=\0    J 

5.  Clear,  the  1st  of  frac.  t/+2z+to-=16  ^ 

6.  Subtract.  2d  from  3d,  z-w  =  S>  Three  equations 

7.  Subtract.  4th  from  3d,  y  -w  =  2) 

8.  Adding  5th  and  6th,  y+Sz=l9  >  ^      eauations 

9.  Subtract.  7th  from  6th,       -i/+^=l  ]  ^""^  equations. 

10.  Adding  8th  and  9th,  4z=20.     Or  z=5  \ 

11.  Transp.  in  the  8th,  y  =  \9  -Sz=4        f  Quantities 

12.  Transp.  in  the  3d,  x=\2-y-z=S     i    required. 

13.  Transp.  in  the  2d,  w=9-x-y=2      3 

r  w-\-50=x      ^ 

Prob.  24.  Given  )  "^JlllQ^gz  [  ^^  ^^^  *^'  ^'  ^'  ^^^  ** 
i  z-{-lQ5  =  Sw  ) 
Answer.  w  =  100  !/  =  90 

a:=150  ^=105. 

Prob.  25.  There  is  a  certain  number  consisting  of  two 
digits.  The  left-hand  digit  is  equal  to  3  times  the  right- 
hand  digit;  and  if  twelve  be  subtracted  from  the  number 
itself,  the  remainder  will  be  equal  to  the  square  of  the  left- 
hand  digit.     What  is  the  number  1 

Let  x=  the  left-hand  digit,  and  1/=  the  right  hand  digit. 

As  the  local  value  of  figures  increases  in  a  ten-fold  ratio 
from  right  to  left ;  the  number  required  =lOx-\-y 

By  the  conditions  of  the  problem  x=Sy  ) 

And  10a:+2/-12  =  a^  J 

The  required  number  is,  therefore,  93. 


EQUATIONS.  165 

Prob.  20.  If  a  certain  number  be  divided  by  the  product 
of  its  two  digits,  the  quotient  will  be  2  ;  and  if  27  be  added 
to  the  number,  the  d'gits  will  be  inverted.  What  is  the 
number]  Ans.  36. 

Prob.  27.  There  are  two  numbers,  such,  that  if  the  less  be 
taken  from  three  times  the  greater,  the  remainder  will  be  35  ; 
and  if  4  times  the  greater  be  divided  by  3  times  the  less  -\-\, 
the  quotient  will  be  equal  to  the  less.  What  are  the  numbers  1 

Ans.  13  and  4. 

Prob.  28.  There  is  a  certain  fraction,  such,  that  if  3  be 
added  to  the  numerator,  the  value  of  the  fraction  will  be  i  ; 
but  if  1  be  subtracted  from  the  denominator,  the  value  will 
be  i.     What  is  the  fraction  ?  »        4 

^'2r 

Prob.  29.  A  gentleman  has  two  horses,  and  a  saddle  wiiich 
is  worth  ten  guineas.  If  the  saddle  be  put  on  the  first  horse, 
the  value  of  both  will  be  double  that  of  the  second  horse  ;  but 
if  the  saddle  be  put  on  the  second  horse,  the  value  of  both 
will  be  less  than  that  of  the  first  horse  by  13  guineas.  What 
is  the  value  of  each  horse  1 

Ans.  56  and  33  guineas. 

Prob.  30.  Divide  the  number  90  into  4  such  parts,  that  the 
first  increased  by  2,  the  second  diminished  by  2,  the  third  mul- 
tiplied by  2,  and  the  fourth  divided  by  2,  shall  all  be  equal. 

If  X,  y,  and  z,  be  three  of  the  parts,  the  fourth  will  be 
90  -x-y-z.     And  by  the  conditions, 

x-\-2=y-2 
x-\-2  =  2z 

o._90-a;-y-z 


The  parts  required  are  18,  22,  10,  and  40. 

Prob.  31.  Find  three  nmiabers,  such  that  the  first  with  I 
the  sum  of  the  second  and  third  shall  be  120  ;  the  second  with 
r  the  difference  of  the  third  and  first  shall  be  70  ;  and  5  the 
sum  of  the  three  numbers  shall  be  95. 

Prob.  32.  What  two  numbers  are  those,  whose  difference, 
sum  and  product,  are  as  the  numbers  2,  3,  and  5  1 

Ans.  10  and  2. 


166  ALGEBRA. 

Prob  33.  A  vintner  sold  at  one  time,  20  dozen  of  port 
wine,  and  30  dozen  of  sherry  ;  and  for  l.he  whole  received 
120  guineas.  At  another  time,  he  sold  30  dozen  of  port  and 
25  dozen  of  sherry,  at  the  same  prices  as  before  ;  and  for  the 
whole  received  140  guineas.  What  was  the  price  of  a  dozen 
of  each  sort  of  wine  ] 

Ans.  The  port  was  3  guineas,  and  the  sherry  2  guineas  a 
dozen. 

Prob.  34.  A  merchant  having  mixed  a  certain  number  of 
gallons  of  brandy  and  water,  found  that,  if  he  had  mixed  6 
gallons  more  of  each,  he  would  have  put  into  the  mixture  7 
gallons  of  brandy  for  every  6  of  water.  But  if  he  had  mixed 
6  less  of  each,  he  would  have  put  in  6  gallons  of  brandy  for 
every  5  of  Avater.  How  many  gallons  of  each  did  he  mix? 
Ans.  78  gallons  of  brandy  and  66  of  water. 

Prob.  35.  What  fraction  is  that,  whose  numerator  being 
doubled,  and  the  denominator  increased  by  7,  'the  value  be- 
comes f  ;  but  the  denominator  being  doubled,  and  the  nume- 
rator increased  by  2,  the  value  becomes  f'?  Ans.  ^. 

Prob.  36.  A  person  expends  30  cents  in  apples  and  pears, 
giving  a  cent  for  4  apples  and  a  cent  for  5  pears.  He  after- 
wards parts  Avith  half  his  apples  and  one  third  of  his  pears, 
the  cost  of  which  was  13  cents.  Hoav  many  did  he  buy  of 
each  ]  Ans.  72  apples  and  60  pears.* 


335.  If  in  the  algebraic  statement  of  the  conditions  of  a 
problem,  the  original  equations  are  more  numerous  than  the 
unknown  quantities  ;  these  equations  will  either  be  contra- 
dictory, or  one  or  moie  of  them  will  be  superfluous. 

Thus  the  equations      <   r,~c>r|  [     are  contradictory. 

For  by  the  first  a: =20,  while  by  the  second,  a:  =  40. 
\But  if  the  latter  be  altered,  so  as  to  give  to  x  the  same  value 
as   the   former,  it  Avill   be  useless,  in  the   statement  of   a 


+  For  more  examples  of  the  solution  of  problems  by  ec^uations,  see  Euler's 
Algebi-a,  Part  I,  Sec.  4  ;  Simpson's  Algebra,  Sec.  II  ;  Smipson's  Exercises  ; 
Maclaurin's  Algebra,  Part  I,  Chap.  2  and  13  ;  Emerson's  Algebra,  Book  II, 
Sec.  I ;  Saunderson's  Algebra,  Book  II  and  III ;  Dodson's  Mathematical  Re- 
pository, and  Bland's  Algebraical  Problems. 


EQUATIONS.  167 

problem.     For  nothing  can   be  determined   from   the  one, 
which  cannot  be  from  the  other. 

Thus  of  the  equations    <  i'^Zin  (  *^^^^  ^^  superfluous. 

For  either  of  tliem  is  sufficient  to  determine  the  vahte  of  x. 
They  are  not  independent  equations.  (Art.  322.)  One  is 
convertible  into  the  other.  For  if  we  divide  the  1st  by  6,  it 
will  become  the  same  as  the  second. 

Or  if  we  multiply  the  second  by  6,  it  will  become  tlie  same 
as  the  first. 

336.  But  if  the  number  of  hidependent  equations  produc- 
ed from  the  conditions  of  a  problem,  is  less  than  the  number 
of  unknown  quantities,  the  subject  is  not  sufficiently  limited 
to  admit  of  a  definite  answer.  For  each  equation  can  limit 
but  one  quantity.  And  to  enable  us  to  find  this  quantity,  all 
the  others  connected  with  it,  must  either  be  previously  known, 
or  be  determined  from  other  equations.  If  this  is  not  the 
case,  there  will  be  a  variety  of  answers  which  Avill  equally 
satisfy  the  conditions  of  the  question.  If,  for  instance,  in 
the  equation 

xJry=lOO, 
X  and  y  are  required,  there  may  be  fifty  different  answers. 
The  values  of  x  and  y  may  be  either  99  and  1,  or  98  and  2, 
or  97  and  3,  &c.  For  the  sum  of  each  of  these  pairs  of 
numbers  is  equal  to  100.  But  if  there  is  a  second  equation 
which  determines  one  of  these  quantities,  the  other  may  then 
be  found  from  the  equation  already  given.  As  x-\-y=\QO, 
if  a:=46,  y  must  be  such  a  number  as  added  to  46  will  make 
100,  that  is,  it  must  be  54.  No  other  number  will  answer 
this  condition. 

337.  For  the  sake  of  abridging  the  solution  of  a  problem, 
however,  the  number  of  independent  equations  actually  put 
upon  paper  is  frequently  less,  than  the  number  of  unknown 
quantities.  Suppose  we  are  required  to  divide  100  into  two 
such  parts,  that  the  greater  shall  be  equal  to  three  times  the 
less.  If  Vv^c  put  X  for  the  greater,  the  less  will  be  100 -a,-. 
(Art.  195.) 

Then  by  the  supposition,  x  —  ZOO  -  3j. 

Transposing  and  dividing,  x—7o,  the  greater. 

And  100-  75  =  25,  the  less. 


168  ALGEBRA. 

Here,  two  unknown  quantities  are  found,  although  there 
appears  to  be  but  one  independent  equation.  The  reason  of 
this  is,  that  a  part  of  the  sokuion  has  been  omitted,  because 
it  is  so  simple,  as  to  be  easily  supplied  by  the  mind.  To 
have  a  view  of  the  whole,  without  abridging,  let  x=  the 
greater  number,  and  y=  the  less. 

1.  Then  by  the  supposition,  x-\-y=\00  } 

2.  And  Sy  =  x  5 

3.  Transposing  x  in  the  1st,  y=\00-x 

4.  Dividing  the  2d  by  3,  y=lx 

6.  Making  the  3d  and  4th  equal,         ^t=100-  x 

6.  Multiplying  by  3,  a— 300  -  Sx 

7.  Transposing  and  dividhig,  x='75,  the  greater. 

8.  By  the  3d  step,  y=lOO-x=2o,  the  less. 

By  comparing  these  two  solutions  with  each  other,  it  will 
be  seen  that  the  first  begins  at  the  6th  step  of  the  latter,  all 
the  preceding  parts  being  omitted,  because  they  are  too  sim- 
ple to  require  the  formality  of  writing  down. 

Prob.  To  find  two  numbers  whose  sum  is  30,  and  the  dif- 
ference of  their  squares  120. 

Leta=30  6=120 

x=  the  less  number  required. 
Then  a-x=  the  greater.      (Art.  195.) 
And  a^-'2ax-\-x'^=  the  square  of  the  greater.      (Art.  214.) 
From  this  subtract  x%  the  square  of  the  less,  and  we  shall 
have  a^-2ax=  the  difference  of  their  squares. 

Therefore,  .='^=(32)!.-_liO=13. 
2a  2x30 

338.  In  most  cases  also,  the  solution  of  a  problem  which 
contains  many  unkncKvn  quantities,  may  be  abridged,  by  par- 
ticular artifices  in  substituting  a  single  letter  for  several. 
(Art.  321.) 

*  Suppose  four  numbers,  u,  x,  y  and  z,  are  required,  of  which 
The  sum  of  the  three  first  is  IS 

The  sum  of  the  two  first  and  last  17 

The  sum  of  the  first  and  two  last  1 8 

The  sum  of  the  three  last  21 


Ludlam's  Algebra,  Art.  161.  c. 


EQUATIONS.  ie9 

Then  1.  u-{-x-{-yz=z  IS 

2.  u-\-x-^z=l7 

3.  u-\-y-\-z=\8 

4.  x-^yJr^=2\. 

Let  S  be  substituted  for  the  sum  of  the  four  numbers,  that 
is,  for  u-^x-{-y-]-z.  It  will  be  seen  that  of  these  four  equa- 
tions. 

The  first  contains  all  the  letters  except  z,  that  is,  S-z=  13 
The  second  contains  all  except  i/,  that  is,  S-y=\7 

The  third  contains  all  except  x,  that  is,  *S*-  x=\S 

The  fourth  contams  all  except  w,  that  is  *S'-  u=21. 

Adding  all  these  equations  together,  we  have 

4S -z-  y  -X  -u=G9 
Or  4^-  (--f?/+a:+w)  =  69  (Art.  88.  c.) 
But  S=(z-}-y-{-x-\-u)  by  substitution. 
Therefore,  4>S*-;S'=69,  that  is,  SS=69,  and  S=z2S, 

Then  putting  23  for  S,  in  the  four  equations  in  which  it 
is  first  introduced,  we  have 

23-2:^13]  f2:r.23  -13=10 

23-2/  =  17l^j  U  =  23-17  =  6 

23-a:=18  j^  ^^^eieioic^j  ^,^23-18  =  5 

23-it  =  21J  [1^^23-21  =  2. 

Contrivances  of  this  sort  for  facilitating  the  solution  of 
particular  problems,  must  be  left  to  be  furnished  for  the  occa- 
sion, by  the  ingenuity  of  the  learner.  They  are  of  a  nature 
not  to  be  taught  by  a  system  of  rules. 

339.  In  the  resolution  of  equations  containing  several  un- 
known quantities,  there  w411  often  be  an  advantage  in  adopt- 
ing the  following  method  of  notation. 

The  co-efficients  of  one  of  the  unknown  quantities  are 
represented, 

In  the  first  equation,  by  a  single  letter,  as  a. . 

In  the  second,  by  the  same  letter  marked  with  an  accent,  as  a^. 

In  the  third,  by  the  same  letter  with  a  double  accent,  as  a^^,&c. 

The  co-efficients  of  the  other  unknown  quantities,  are  re- 
presented by  other  letters  marked  in  a  similar  manner  ;  as  are 
also  the  terms  which  consist  of  known  quantities  onl}'. 

22 


170  ALGEBRA. 

Two  equations  containing  the  two  unknown  quantities  x 
and  y  may  be  written  thus, 

ax-\-hy=c 

Three  equations  containing  x,  y,  and  z,  thus, 

ax-4-by-\-cz=d 

a'x-\-h'y-\-&z=d' 

a''x-{-h''y+c''z=d'\ 

Four  equations  containing  x,  i/,  z,  and  u,  thus, 

ax-\-by-\-cz-\-du=:e 
a'x-\-h'y^&z-\-d'u=e' 
a''x-\-  b''y-{-c''zJf-  d''u = e" 
a'''x-{-  b'''y-{-c'''z-{-  d'^'u = e'''. 

The  same  letter  is  made  the  co-efficient  of  the  same  un- 
known quantity,  in  different  equations,  that  the  co-efficients 
of  the  several  unknown  quantities  may  be  distinguished,  in 
any  part  of  the  calculation.  But  the  letter  is  marked  with 
different  accents,  because  it  actually  stands  for  different  quan- 
tities. 

Thus  we  mayputa=4,  a'=:6,  a'' =  10,  a'-"  =  20,  &c. 
To  find  the  value  of  x  and  y. 

1.  In  the  equation,  ax-^by  =  c     ) 

2.  And  .  a'x^b'y  =  c'i 

3.  Multiplying  the  1st  by  6^,(Art.  S2S.) ab'x-]-bb'y  =  cb' 

4.  Multiplying  the  2d  by  b,  ba'x-\-bb'y  =  bc^ 

5.  Subtracting  the  4th  from  the  3d,       a¥x  -  ba'x=cV  -  h& 

6.  Dividing  by  aW -ba\  (Art.  121.) 
By  a  similar  process,  y  = 

The  symmetry  of  these  expressions  is  well  calculated  to  fix 
them  in  the  memory.  The  denominators  are  the  same  in 
both  ;  and  the  numerators  are  like  tlie  denominators,  except 
a  change  of  one  of  the  letters  in  each  terin.  But  tlie  par- 
ticular advantage  of  this  method  is,  that  the  expressions  here 
obtained  may  be  considered  as  general  solutions,  which  give 
the  values  of  ihe  unknow^n  quantities,  in  other  equations,  of 
a  similar  nature. 


EQUATIONS.  171 

Thus  if  10j;+6y=100  ) 

And        40x-{-4y=200l 

Then  putting  rt=:  10  b=6  c=100 

a'=40  b'=4  c'=200 

^    ,  cb'-bc'     100x4-6x200      . 

We  have  x=— = — ~ =4. 

ab'-ba'       10x4-6x40 

4nd  ag^-m^_  10x200- 100X40  _^^ 

ab'-ba'  10x4-6x40 

The  equations  to  be  resolved  may,  originally,  consist  of 
more  than  three  terms.  But  if  they  are  of  the  first  degree, 
and  have  only  two  unknown  quantities,  each  ma}'  be  reduced 
to  three  terms  by  substitution. 

Thus  the  equation  dx  -  4x-\-hy  -  6y=m-\-8 

Is  the  same,  by  Art.  120,  as  {d-4)x-{-{h-6)y=m-{-S. 
And  putting     a=zf/-4,  6  =  /i-6,  c  —  m-\-S 

It  becomes  ax-\-by=c.* 


DEiMONSTRATION  OF  THEOREIMS. 

340.  Equations  have  been  applied,  in  this  and  the  preced- 
ing sections,  to  the  solution  of  problems.  They  may  be  em- 
ployed with  equal  advantage,  in  the  demonstration  of  theo- 
rems. The  principal  diflerence,  in  the  two  cases,  is  in  the 
order  in  which  the  steps  are  arranged.  The  operations  them- 
selves are  substantially  the  same.  It  is  essential  to  a  demon- 
stration, that  complete  certainty  be  carried  through  every 
part  of  the  process.  (Art.  11.)  This  is  effected,  in  the  re 
duction  of  equations,  by  adhering  to  the  general  rule,  to  make 
no  alteration  which  shall  allect  the  value  of  one  of  the  mem- 
bers, without  equally  increasing  or  diminishing  the  other. 
In  applying  this  principle,  we  are  guided  by  the  axioms  laid 
down  in  Art.  63.  These  axioms  are  as  applicable  to  the  de- 
monstration of  theorems,  as  to  the  solution  of  problems. 

But  the  order  of  the  stops  vdll  generally  be  different.  In 
solving  a  problem,  the  object  is  to  find  the  value  of  the  un- 
known quantity,  by  disengaging  it  from  all  other  quantities. 
But,  in  conducting  a  demonstration,  it  is  necessary  to  bring 

*  For  the  application  of  this  plan  of  notation  to  the  solution  of  equations 
which  contain  more  than  two  unknown  quantities,  see  LaCroix's  Algebra,  Art. 
85  ;  Maclaurin's  Algebra,  Part.  I.  Chap.  12  ;  Fenn's  Algebra,  p.  57  ;  and  a 
paper  of  Laplat^,  in  the  ?>lemoiri3  of  th«  Acactemy  of  F^ciences  for  1772. 


172  ALGEBRA. 

the  equation  to  tliat  particular  form  which  will  express,  in 
algebraic  terms,  the  proposition  to  be  proved. 

Ex.  1.  Theorem.  Four  times  the  product  of  any  two 
numbers,  is  equal  to  the  square  of  their  sum,  diminished  by 
the  square  of  their  difference. 

Let  rr=  the  greater  number,  s—  their  sum, 

V=  the  less,  d=i  their  difference. 

Demonstration. 

1.  By  the  notation  x-\-y  =  s    > 

2.  And  x-y  =  d    ] 

3.  Adding  the  two,  (Ax.  1.)  '^x^s-^-d 

4.  Subtracting  the  2d  from  the  1st,  2y  =  s-d 

5.  Mult.  3d  and  4th,  (Ax.  3.)  4xyz=(s-lrd)x{s-d) 

6.  That  is,  (Art.  235.)  4xy=s^-'d:' 

The  last  equation  expressed  in  words  is  the  proposition 
which  w^as  to  be  demonstrated.  It  will  be  easily  seen  that 
it  is  equally  applicable  to  any  two  numbers  whatever.  For 
the  particular  values  of  x  and  y  will  make  no  difference  in 
the  nature  of  the  proof. 

Thus  4x8x6=(8+6)-^- (8-6)^=192. 
And  4xl0x6=(104-6)2- (10-6)^^=240. 
And  4xl2xl0=(12+10)-- (12- 10)2=480. 

Theorem  2.  The  sum  of  the  squares  of  any  two  numbers  is 
equal  to  the  square  of  their  difference,  added  to  twice  their 
product. 

Let  x=  the  greater,  dz=  their  difference. 

y=  the  less,  p=  their  product. 

Demonstration.  • 

1 .  By  the  notation  x-y  =  d 

2.  And  xy=p 

3.  Squaring  the  first  x'^-2xy+y'^^d^ 

4.  Multiplying  the  second  by  2  2xyz=2p 

5.  Adding  the  third  and  fourth  x'^-j-y'^=d^--\-2p. 

Thus  10*-f82=(10-8)24-2xl0x8=164. 

341.  General  propositions  are  also  discovered,  in  an  expedi- 
tious manner,  by  means  of  equations.  The  relations  of 
quantities  may  be  presented  to  our  view,  in  a  great  variety 


RATIO.  173 

of  Avays  by  the  several  changes  through  which  a  given  equa- 
tion maybe  made  to  pass.  Eadi  step  in  tlie  process  will 
contain  a  distinct  proposition. 

Let  s  and  d  be  the  sum  and  difference  of  two  quantities  x 
and  y,  as  before. 

1.  Then  s=x-\-y) 

2.  And  d=x-y) 

3.  Dividing  the  first  by  2,  J^^i^+il/ 

4.  Dividing  the  2d  by  2,  ^^(^zzz^x-iy 

5.  Adding  the  3d  and  4th,  is-\-hd=lx-\-lx=x 

6.  Sub.  the  4th  from  the  3d,         hs-id=iy-{-^=y. 

That  is, 

Half  the  difference  of  Uco  quantities,  added  to  half  their  sum,  is 
equal  to  the  greater ;  and 

Half  their  difference  subtracted  from  half  their  sum,  is  equal  to 
the  less. 


SECTION  XII. 


RATIO  AND  PROPORTION.* 


Art.  342.  THE  design  of  mathematical  investigations,  is 
to  arrive  at  the  knowledge  of  particular  quantities,  by  com- 
paring them  with  other  quantities,  either  equal  to,  or  greater 
or  less  than  those  which  are  the  objects  of  inquiry.    The  end 


♦  Euclid's  Elements,  Book  5,  7,  8.  Euler's  Algebra,  Part  I.  Sec.  3.  Emerson 
on  Proportion.  Camus' Geometry,  Book  III.  Ludlam's  Mathematics.  Walks' 
Alffebra  Chap.  19,  20.  Saunderson's  Algebra,  Book  7.  Barrow's  Mathema- 
tical Lectures.  Analyst  for  March,  1814.  Port  Royal  Art  of  Thinking,  Part 
IV.  Ch.  iv. 


174  ALGEBRA. 

is  most  commonly  attained  by  means  of  a  series  of  equations 
and  proportions.  When  we  make  use  of  equations,  we  deter- 
mine the  quantity  sought,  by  discovering  its  equality  with 
some  other  quantity  or  quantities  aheady  known. 

We  have  frequent  occasion,  however,  to  compare  the  un- 
known quantit}'  witli  others  which  are  not  equal  to  it,  but 
either  greater  or  less.  Here  a  different  mode  of  proceeding 
becomes  necessary.  We  may  inquire,  either  Jiow  much  one 
of  the  quantities  is  greater  than  the  other ;  or  how  many  times 
the  one  contains  the  other.  In  finding  the  answer  to  either 
of  these  inquiries,  we  discover  what  is  termed  a  ratio  of  the 
two  quantities.  One  is  called  arithmetical  and  the  other  geo- 
metrical ratio.  It  should  be  observed,  however,  that  both 
these  terms  have  been  adopted  arbitrarily,  merely  for  dis- 
tinction's sake.  Arithmetical  ratio,  and  geometrical  ratio  are 
both  of  them  applicable  to  arithmetic,  and  both  to  geometry. 

As  the  whole  of  the  extensive  and  important  subject  of  pro- 
portion depends  upon  ratios,  it  is  necessary  that  these  should 
be  clearly  and  fully  understood. 

343.  Arithmetical  ratio  is  the  difference  between  two 
quantities  or  sets  of  quantities.  The  quantities  themselves  are 
called  the  tenns  of  the  ratio,  that  is,  the  terms  between  which 
the  ratio  exists.  Thus  2  is  the  arithmetical  ratio  of  5  to  3. 
This  is  sometimes  expressed,  by  placing  two  points  between 
the  quantities  thus,  5. .  3,  which  is  the  same  as  5  -3.  Indeed 
the  term  arithmetical  ratio,  and  its  notation  by  points,  are 
almost  needless.  For  the  one  is  only  a  substitute  for  the  word 
difference,  and  the  other  for  the  sign  -. 

344.  If  both  the  terms  of  an  arithmetical  ratio  be  multiplied 
or  divided  by  the  same  quantity,  the  ratio  will,  in  effect,  be 
multiplied  or  divided  by  that  quantity. 

Thus  if  a-b=r 

Then  mult,  both  sides  by  h,  (Ax.  3.)       ha-hh=hr 

a     o      r 
And  dividing  by  h,  (Ax.  4.)  ^  -  j'=j^ 

345.  If  the  terms  of  one  arithmetical  ratio  be  added  to,  or 
subtracted  from,  the  corresponding  terms  of  another,  the  ratio 
of  their  sum  or  difference  will  be  equal  to  the  sum  or  differ- 
ence of  the  two  ratios. 


RATIO.  175 

If      a-b)         ,,     , 

And  d-h  r'^ '^'^ '''''' ''''''''^ 
Then  {aJrd)-{bJrh)  =  {a-b)J^{d-h).  For  each  =a-\-fi-b-Iu 
And  (a-d)-{b-h)  =  {a-b)-{d-h).  For  each  =a-d-bJ[-h. 
Thus  the  arith.  ratio  of  1 1 . .  4  is  7  / 
And  the  arith.  ratio  of      5.. 2  is  3  3 
The  ratio  of  the  sum  of  the  terms  16.  .6  is  10,  the  sum  of 

the  ratios. 
The  ratio  of  the  difference  of  tlie  terms  6.  .2  is  4,  the  differ- 
ence of  the  ratios. 

346.  GEOMETRICAL  RATIO  is  that  relatio>'  be- 
tween QUANTITIES  WHICH  IS  EXPRESSED  BY  THE  QUO- 
TIENT   OF    THE    ONE    DIVIDED    BY    THE    OTHER.* 

Tiius  the  ratio  of  8  to  4,  is  f  or  2.  For  this  is  the  quotient 
of  8  divided  by  4.  In  other  words,  it  shows  how  often  4  is 
contained  in  8. 

In  the  same  manner,  the  ratio  of  any  quantity  to  another 
may  be  expressed  by  dividing  the  former  by  the  latter,  or, 
which  is  the  same  thing,  making  the  former  the  numerator 
of  a  fraction,  and  the  latter  the  denominator. 

a 
Thus  the  ratio  of  a  to  b  is  t' 

d-\-h 
The  ratio  of  d-{-h  to  b-\-c,  is  ,  .    • 

347.  Geometrical  ratio  is  also  expressed  by  placing  two 
points,  one  over  the  other,  between  the  quantities  compared. 

Thus  a  :  b  expresses  the  ratio  of  a  to  6 ;  and  12:4  the  ratio 
of  12  to  4.  The  two  quantities  together  are  called  a  couplet, 
of  which  the  first  term  is  the  antecedent,  and  the  last,  the 
consequent. 

348.  This  notation  by  points,  and  the  other  in  the  form  of 
a  fraction,  may  be  exchanged  the  one  for  the  other,  as  con- 
venience may  require  ;  observing  to  make  the  antecedent  of 
the  couplet,  the  numerator  of  the  fraction,  and  the  consequent 
the  denominator. 

h 

Thus  10  :  5  is  the  same  as  ^5-  and  b  :  d,  the  same  as  -r- 

349.  Of  these  three,  the  antecedent,  the  consequent,  and 
the  ratio,  any  two  being  given,  the  other  may  be  found. 

*  See  Note  H. 


176  ALGEBRA. 

Let  a=  the  antecedent,  c=  the  consequent,  r=  the  ratio. 

By  definition  r=-;  that  is,  the  ratio  is  equal  to  th«  antece- 
dent divided  by  the  consequent.  *  > 

Multiplying  by  c,  a=cr,  that  is,  the  antecedent  is  ecfiml  to 
the  consequent  multiplied  into  the  ratio. 

Dividing  by  4*,  c=-,  that  is,  the  consequent  is  equal  to  the 
antecedent  divided  by  the  ratio. 

Cor.  1.  If  two  couplets  have  their  antercedentrf^qual,  and 
their  consequents  equal,  their  ratios  must  be  equal.* 

Cor.  2.  If,  in  two  couplets,  the  ratios  are  equal,  and  the 
antecedents  equal,  the  consequents  are  equal ;  and  if  the 
ratios  are  equal  and  the  consequents  equal,  the  antecedents 
are  equal,  f 

350.  If  the  two  quantities  compared  are  equal,  the  ratio  is 
a  unit,  or  a  ratio  of  equality.  The  ratio  of  3x6  :  18  is  a 
unit,  for  the  quotient  of  any  quantity  divided  by  itself  is  1 . 

If  the  antecedent  of  a  couplet  is  greater  than  the  conse- 
quent, the  ratio  is  greater  than  a  unit.  For  if  a  dividend  is 
greater  than  its  divisor,  the  quotient  is  greater  than  a  unit. 
Tfius  the  ratio  of  18  :  6  is  3.  (Art.  128.  cor.)  This  is  called 
a  ratio  of  greater  inequality. 

On  the  other  hand,  if  *he  antecedent  is  less  than  the  con- 
sequent, the  ratio  is  less  tlian  a  unit,  and  is  called  a  ratio  of 
less  ineqiiality.  Thus  the  ratio  of  2:3,  is  less  than  a  unit, 
because  the  dividend  is^  less  than  the  divisor. 

351.  INVERSE  or  RECIPROCAL  ratio  is  the  ratio 

OF  THE   RECIPROCALS  OF  TWO  QUANTITIES.       See  Art.  49. 

Thift  the  reciprocal  ratio        6^ to  3,  is  i  to  i,  that  is  -e-r-i* 

ft 
The  difect  ratio  of  a  to  6  is  T,th..t  is,  the  antecedent  divided 

by  the  consequent. 

I     I      ^      ^      I      b      h 

The  reciprocal  ratio  is  -  *  ror  --7-t=i-Xt  ==~  • 
^  a    b      a     b     a     \      a 

that  is  the  consequent  h  divided  hy  the  antecedent  a. 


*  Euclid  7.  5.  t  ^uc.  9.  5. 


RATIO.  177 

Hence  a  reciprocal  ratio  is  expressed  by  inverting  the  frac^ 
tton  which  expresses  the  direct  ratio  ;  or  when  the  notation 
is  by  points,  by  inverting  the  order  of  the  terms. 

Thus  a  is  to  6,  inversely,  as  b  to  a. 

352.  COMPOUND  RATIO  is  the  ratio  of  the  PRO- 
DUCTS, OF  THE  CORRESPONDING  TERMS  OF  TWO  OR  MORE 
SIMPLE    RATIOS.* 

Thus  the  ratio  of  6  :  3,  is  2 

And  the  ratio  of  12  :  4,  is  3 


The  ratio  compounded  of  these  is  72  :  12  =  6. 

Here  the  compound  ratio  is  obtained  by  multiplying 
together  the  two  antecedents,  and  also  the  two  consequents, 
of  the  simple  ratios. 

So  the  ratio  compounded, 

Of  the  ratio  of  a  :  b 

And  the  ratio  of  c  .  d 

And  the  ratio  of  h  :  y 

Is  the  ratiiJ  of  ach  :  bdy  —  ~~ 

bdy 

Compound  ratio  is  not  different  in  its  nature  from  any  other 
ratio.  The  term  is  used,  to  denote  the  origin  of  the  ratio,  in- 
particular  cases. 

Cor.  The  compound  ratio  is  equal  to  the  product  of  the 
simple  ratios. 

The  ratio  of  a  :  b  is  -. 

0 

The  ratio  of  c  :  d,  is  - 

d 

The  ratio  of  h  :  y,  is  - 

y 

And  the  ratio  compounded  of  these  is  — ,   which    is    the 

bdy 

product  of  the  fractions  expressing  the  simple  ratios.  (Art. 
155.) 

353.  It,  in  a  series  of  ratios,  the  consequent  of  each  pre- 
ceding couplet,  is  the  antecedent  of  the  following  one,  the 


*  See  Note  I. 
28 


178  ALGEBRA. 

ratio  of  the  first  antecedent  to  the  last  consequent^  is  equal  to  that 
which  is  compounded  of  all  the  intervening  ratios.* 
Thus,  in  the  series  of  ratios  a  :  b 

b:c 
c  :  d 
d:  h 
the  ratio  of  a  :  h  is  equal  to  that  which  is  compounded  of  the 
ratios  of  a  :  b,  of  b  :  c,  of  c  :  d,  of  d  :  h.     For  the  compound 

ratio  by  the  last  article  is  ^ =-  or  a  :  h.   (Art.  145.) 

bcdh     h 

In  the  same  manner,  all  the  quantities  which  are  both 
antecedents  and  consequents  will  disappear  when  the  frac- 
tional product  is  reduced  to  its  lowest  terms,  and  will  leave 
the  compound  ratio  to  be  expressed  by  the  first  antecedent 
and  the  last  consequent. 

354.  A  particular  class  of  compound  ratios  is  produced,  by 
multiplying  a  simple  ratio  into  itself  or  into  another  equal 
ratio.  These  are  termed  duplicate,  triplicate,  quadruplicate, 
&c.  according  to  the  number  of  multiplications. 

A  ratio  compounded  of  two  equal  ratios,  that  is,  the  square 
of  the  simple  ratio,  is  called  a  duplicate  ratio. 

One  compounded  of  three,  that  is,  the  cube  of  the  simple 
ratio,  is  called  triplicate,  &c. 

In  a  similar  manner,  the  ratio  of  the  square  roots  of  two 
quantities,  is  called  a  subduplicate  ratio  ;  that  of  the  cube 
roots  a  subtripUcate  ratio,  &c. 

Thus  the  simple  ratio  of  a  to  b,  is  a  :  b 
The  duphcate  ratio  of  a  to  b,  is  a^  :  b^ 
The  triplicate  ratio  of  a  to  b,  is  a^  :  b^ 
The  subduplicate  ratio  of  a  to  b,  is  \/a  :  \/b 
The  subtriplicate  of  a  to  b,  is  ^a  :  ^yb,  &c. 
The  terms  duplicate,  triplicate,  &c.  ought  not  to  be  con- 
founded with  double,  triple,  &c.f 

The  ratio  of  6  to  2  is  6  :  2  =  3 

Double  this  ratio,  that  is,  twice  the  ratio,  is    12:2  =  6 


Triple  the  ratio,  i.  e.  three  times  the  ratio,  is    18:2 


:si 


*  This  is  the  particular  case  of  compound  ratio  which  is  treated  of  in  the 
5th  book  of  Eucli«l.     See  the  editioas  of  Sirnson  and  Playfaij-. 
t  See  Note  K. 


RATIO.  179 

But  the  duplicate  YSiiio,\.e. the  square  of  the  iatio,is  6" :  2'^=9  ) 
And  the  triplicate  ratio,i.e.the  cube  of  the  ratio,  is  6^ :  2^=27  i 

355.  That  quantities  may  have  a  ratio  to  each  other,  it  is 
necessary  that  they  should  be  so  far  of  the  same  nature,  as 
that  one  can  properly  be  said  to  be  either  equal  to,  or  greater, 
or  less  than  the  other.  A  foot  has  a  ratio  to  an  inch,  for  one 
is  twelve  times  as  great  as  the  other.  But  it  cannot  be  said 
that  an  hour  is  either  shorter  or  longer  tlian  a  rod  ;  or  that 
an  acre  is  greater  or  less  than  a  degree.  Still  if  these  quan- 
tities are  expressed  by  numbers,  there  may  be  a  ratio  between 
the  numbers.  There  is  a  ratio  between  the  number  of  min- 
utes in  an  hour,  and  the  number  of  rods  in  a  mile. 

356.  Having  attended  to  the  nature  of  ratios,  we  have  next 
to  consider  in  what  manner  they  v/ill  be  alTected,  by  varying 
one  or  both  of  the  terms  between  which  the  comparison  is 
made.  It  must  be  kept  in  mind  that,  when  a  direct  ratio  is 
expressed  by  a  fraction,  the  antecedent  of  the  couplet  is  always 
the  numerator,  and  the  consequent  the  denominator.  It  will 
be  easy,  then,  to  derive  from  the  properties  of  fractions,  the 
changes  produced  in  ratios  by  variations  in  the  quantities 
compared.  For  the  ratio  of  the  two  quantities  is  the  same  as 
the  value  of  the  fractions,  each  being  the  quotient  of  the 
numerator  divided  by  the  denominator.  (Arts.  135,  346.) 
Now  it  has  been  shown,  (Art.  137,)  that  multiplying  the 
numerator  of  a  fraction  by  any  quantity,  is  multiplying  the 
value  by  that  quantity  ;  and  that  dividing  the  numerator  is 
dividing  the  value.     Hence, 

357.  Multiplying  the  antecedent  of  a  couplet  by  any  quantity, 
is  multiplying  the  ratio  by  that  quantity ;  and  dividing  the  ante- 
cedent is  dividing  the  ratio. 

Thus  the  ratio  of         6  :  2  is  3 
And  the  ratio  of         24  :  2  is  12. 

Here  the  antecedent  and  the  ratio,  in  the  last  couplet,  are 
each  four  times  as  great  as  in  the  first. 

The  ratio  of  a  :  b  is         _ 
b 

And  the  ratio  of  na  :  6  is  — 
b  * 


180  ALGEBRA. 

Cor.  With  a  given  consequent,  the  greater  the  antecedenty 
the  greater  the  ratio  ;  and  on  the  other  hand,  the  greater  the 
ratio,  the  greater  the  antecedent.*  See  Art.  137.  cor. 

358.  Multiplying  the  consequent  of  a  couplet  by  any  quantity 
is,  in  effect,  dividing  the  ratio  by  that  quantity  ;  and  dividing  the 
consequent  is  multiplying  the  ratio.  For  multiplying  the  denom- 
inator of  a  fraction,  is  dividing  the  value  ;  and  dividing  the 
denominator  is  multiplying  the  value.     (Art.  138.) 

Thus  the  ratio  of  1 2  :  2,  is  6 
And  the  ratio  of    12  :  4,  is  3. 

Here  the  consequent  in  the  second  couplet,  is  tioice  as  great, 
and  the  ratio  only  half  as  great,  as  in  the  first. 

The  ratio  of  a:  b  is  - 
b 

And  the  ratio  of  a  :  nb,  is  -^ 

nb 

Cor.  With  a  given  antecedent,  the  greater  the  consequent, 
the  less  the  ratio  ;  and  the  greater  the  ratio,  the  less  the  con- 
sequent.! See  Art.  138.  cor. 

359.  From  the  two  last  articles,  it  is  evident  that  multiply- 
ing the  antecedent  of  a  couplet,  by  any  quantity,  will  have  the 
same  effect  on  the  ratio,  as  dividing  the  consequent  by  that 
quantity ;  and  dividing  the  antecedent,  will  have  the  same 
effect  as  multiplying  the  consequent.  See  Art.  139. 

Thus  the  ratio  of  8:4,  is  2 

Mult,  the  antecedent  by  2,  the  ratio  of  16  :  4,  is  4 
Divid.  the  consequent  by  2,  the  ratio  of    8:2,  is  4. 

Cor.  Any  factor  or  divisor  may  be  transferred,  from  the 
antecedent  of  a  couplet  to  the  consequent,  or  from  the  conse- 
quent to  the  antecedent,  without  altering  the  ratio. 

It  must  be  observed  that,  when  a  factor  is  thus  transferred 
ftom  one  term  to  the  other,  it  becomes  a  divisor ;  and  when 
&  divisor  is  transferred,  it  becomes  a  factor. 


Thus  the  ratio  of  3x6:9  =  2 

Transferring  the  factor  3,  6 


:  9=2      , 
:*=2r 


le  same  ratio. 


♦  Euclid  8  and  10.  5.     The  first  part  of  the  propositions, 
t  Euclid  8  and  10.  5.     The  last  part  of  the  propositions. 


RATIO.  1<81 


_,  via  ma  ma 

The  ratio  of  —  :  b= — -^b  = 


y  '        y    '         by 


ma 


Transferring'  y  ma  :  by —  ma~-by  = -j— 


by        ^  by 


ma 


Transferring  m,        a:  —  =  a-^ —  =  , 
°  m         '  m      by 

360.  It  is  farther  evident,  from  Arts.  357  and  358,  that  if 

THE  ANTECEDENT  AND  CONSEQUENT  BE  BOTH  MULTIPLIED, 
OR  BOTH  DIVIDED,  BY  THE  SAME  QUANTITY,  THE  RATIO  WILL 

NOT  BE  ALTERED.*     See  Art.  140. 

Thus  the  ratio  of  8  :  4=2  ^ 

MuU.  both  terms  by  2,    16  :  8  =  2  >  the  same  ratio. 

Divid.  both  terms  by  2,     4:2  =  2) 

a 
The  ratio  of  a:  b=T 

ma     a 
Multiplying  both  terms  by  m,  ma  :  mb  =  —i=T  > 

T^-   .1-       .     1  ,  a    b     an     a 

Dividmar  both  terms  by  n,  -  :  -=7-  =-r 

°  -^     '  n   n     bn     b-J 

Cor.  1 .  The  ratio  of  two  fractions  which  have  a  common 
denominator,  is  the  same  as  the  ratio  of  their  numerators. 

a    b 

Thus  the  ratio  of  - :  -,  is  the  same  as  that  of  a  :  b. 
n   n 

Cor.  2.  The  direct  ratio  of  two  fractions  which  liave  a 
common  numerator,  is  the  same  as  the  reciprocal  ratio  of 
their  denominators. 

a    a  ^  11 

Thus  the  ratio  of  —  ^  -,  is  the  same  as  —  •  -,  or  n  :  m. 
m    n'  m    n' 

361.  From  the  last  article,  it  will  be  easy  to  determine  the 
ratio  of  any  two  fractions.  If  each  term  be  multiplied  by 
the  tw^o  denominators,  the  ratio  will  be  assigned  in  integral 
expressions.  Thus  multiplying  the  terms  of  the  couplet 
a    c  add    bed 

r  •  ^  by  bd,  we  have  —j—  •  —r,  which  becomes  ad  :  be,  by  can- 
celling equal  quantities  from  the  numerators  and  denomi- 
nators. 


*  Eudid.  15.  5 


182  ALGEBRA. 

361.    b.    A  ratio  of  greater  inequality,   compounded  with 
another  ratio,  increases  it. 

Let  the  ratio  of  greater  inequahty  be  that  of  l-\-n  :  1 

And  any  given  ratio,  that  of  a:  b 


The  ratio  compounded  of  these,  (Art.  352,)  is      a-\-na  :  b 
Which  is  greater  than  that  of  a  ;  6  (Art.  356.  cor.) 
But  a  ratio  of  lesser  inequality,  compounded  with  another 
ratio,  diminishes  it. 

Let  the  ratio  of  lesser  inequahty  be  that  of  1  -n  :  1 

And  any  given  ratio,  that  of  a:  b 


The  ratio  compounded  of  these  is  a-na:b 

Which  is  less  than  that  of  a:  b. 

362.  If  to  or  from  the  terms  of  any  cotiplet,  there  be  added  or 
SUBTRACTED  two  Other  quantities  having  the  same  ratio,  the  sums 
or  remainders  loill  also  have  the  same  ratio.* 
Let  the  ratio  of  a  :  b  ) 

Be  the  same  as  that  of  c:  d) 

Then  the  ratio  of  the  sum  of  the  antecedents,  to  the  sum 
of  the  consequents,  viz.  of  a-\-c  to  b-[-d,  is  also  the  same. 
a-\-c      c     a 

Demonstration, 

a     c 

1.  By  supposition,  b~d 

2.  Multiplying  by  b  and  d,  ad=:bc 

3.  Adding  cd  to  both  sides,  ad-{-cdz=zbc-\-cd 

bc-\-cd 

4.  Dividing  by  d,  a-]^c=      ^ 

a-{c      c     a 

5.  Dividing  by  6+rf,  ,  q-^=^=j- 

The  ratio  of  the  difference  of  the  antecedents,  to  the  differ- 
ence of  the  consequents,  is  also  the  same. 
a-  c     c     a 


That 


IS 


b-d-d-b 


*  Euclid,  5  and  6,  5. 


RATIO.  18S 

Demonstration. 


1.  By  supposition,  as  before,  -=:- 

b     d 

2.  Multiplying  by  b  and  d,  ad=bc 

3.  Subtracting  cd  from  both  sides,    ad  -cd=bc-  cd 

4.  Dividing  by  (/,  a-c=^lzS£. 


5.  Dividing  hyb-d  "flZl  =i=t 

b-d     d     b 
Thus  the  ratio  of  15  :  5  is  3  > 

And  the  ratio  of  9  :  3  is  3  3 

Then  adding  and  subtracting  the  terms  of  the  two  couplets, 


The  ratio  of  15+9  :  5+3  is  3 

And  the  ratio  of  15-9:5-3  is  3 


Here  the  terms  of  only  two  couplets  have  been  added  to- 
gether. But  the  proof  may  be  extended  to  any  number  of 
couplets  where  the  ratios  are  equal.  For,  by  the  addition  of 
the  two  first,  a  new  couplet  is  formed,  to  which,  upon  the 
same  principle,  a  third  may  be  added,  a  f  lurib,  &c.    Hence, 

363.  If,  in  several  couplets,  the  ratios  are  equal,  the  sum 

OF  ALL  THE  ANTECEDENTS  HAS  THE  SAME  RATIO  TO  THE 
SUM  OF  ALL  THE  CONSEQUENTS,  WHICH  ANY  ONE  OF  THE 
S  CONSEQ 

12  :  6  =  2 

Thus  the  ratio  i  ^g  ;  \=^ 
6  ':  3  =  2 

Therefore  the  ratioof  (12+10+8+6)  :  (6+5+4+3)  =  2. 

363.  b.  A  ratio  of  greater  inequality  is  diminished,  by  adding 
the  same  quantity  to  both  the  terms. 

Let  the  given  ratio  be  that  of  «+6  ;  a  or  ^it- 

a 

Adding  x  to  both  terms,  it  becomes  a-\-b-\-x  :  a+.r  or  ^~^  Tl^ 

a+ar 


♦Euclid,  1  and  12,  5. 


184  ALGEBRA. 

Reducing  them  to  a  common  denominator, 

The  first  becomes  cr-^ab+ax-Jrbx 

a{a-\-x) 
And  the  latter  a'-^ab+ax 

a(a-^x) 

As  the  latter  numerator  is  manifestly  less  than  the  other, 
the  ratio  must  be  less.     (Art.  356.  cor.) 

But  a  ratio  of  lesser  inequality  is  increased,  by  adding  the 
same  quantity  to  both  terms. 

Let  the  given  ratio  be  that  of  a-  h  :  a,  or  ^~ 

a 

Adding  x  to  both  terms,  it  becomes  a  -  b-\-x  :  a-{-x  or  ^~     '^ 

a-\~x 

Reducing  them  to  a  common  denominator. 

The  first  becomes  n''-ab+ax-b^ 

a(a-\-x) 
And  the  latter,  a'-ab+ax^ 

a(a-\-x) 

As  the  latter  numerator  is  greater  than  the  other,  the  ratio 
is  greater. 

If  the  same  quantity,  instead  of  being  added,  is  subtracted 
from  both  terms,  it  is  evident  the  effect  upon  the  ratio  must 
be  reversed. 

Examples. 

1 .  Which  is  the  greatest,  the  ratio  of  1 1  :  9,  or  that  of 
44  :  35  ? 

2.  Which  is  the  greatest,  the  ratio  of  a-|-3  :  ia,  or  that  of 
2a+7  :ia1 

3.  If  the  antecedent  of  a  couplet  be  65,  and  the  ratio  13, 
what  is  the  consequent  1 

4.  If  the  consequent  of  a  couplet  be  7,  and  the  ratio  18, 
what  is  the  antecedent. 

5.  What  is  the  ratio  compounded  of  the  ratios  of  3  :  7,  and 
2a:  56,  and  7a;+l  :Sy-2l 

6.  What  is  the  ratio  compounded  of  x~\-y  :  b,  and 
x-y  :  a-\-b,  and  a-\-b  :  h  1  Ans.  x"^ - 1/^  :  bh. 


PROPORTION.  185 

7.  If  the  ratios  of  02:+ 7  :2x-S,  and  x-\-2  :  ^  x-\-S  be  com- 

f)bunded,  will  they  produce  a  ratio  of  greater  inequality,  or  of 
esser  inequality  1  Ans.  A  ratio  of  greater  inequality. 

8.  What  is  the  ratio  compounded  of  x-\-y  :  a,  and  x-y  :  b, 

x^  -y~ 
and  b  : 1  Ans.  A  ratio  of  equality. 

9.  What  is  the  ratio  compounded  of  7  :  5,  and  the  dupli- 
cate ratio  of  4  :  9,  and  the  triplicate  ratio  of  3  :  2  1 

Ans.  14:  15. 

10.  What  is  the  ratio  compounded  of  3  :  7,  and  the  tripli- 
cate ratio  of  x  :  y,  and  the  subduplicaie  ratio  of  49  :  9  ? 

Ans.  a;'  :  y'. 

PROPORTION. 

363.  An  accurate  and  familiar  acquaintance  with  the  doc- 
trine of  ratios,  is  necessary  to  a  ready  understanding  of  the 
principles  of  proportion,  one  of  the  most  important  of  all  the 
branches  of  the  mathematics.  In  considering  ratios,  we 
compare  two  quantities,  for  the  purpose  of  finding  either  their 
difference,  or  the  quotient  of  the  one  divided  by  the  other. 
But  in  proportion,  the  comparison  is  between  two  ratios. 
And  this  comparison  is  limited  to  such  ratios  as  are  equal. 
We  do  not  inquire  how  much  one  ratio  is  greater  or  less  than 
another,  but  whether  they  are  the  same.  Thus  the  numbers 
12,  6,  8,  4,  are  said  to  be  proportional,  because  the  ratio  of 
12  :  6  is  the  same  as  that  of  8  :  4. 

364.  Proportion,  then,  is  an  equality  of  ratios.  It  is  ei- 
ther arithmetical  or  geometrical.  Arithmetical  proportion  is 
an  equality  of  arithmetical  ratios,  and  geometrical  proportion 
is  an  equality  of  geometrical  ratios.*  Thus  the  numbers  6, 
4,  10,  8,  are  in  arithmetical  proportion,  because  the  difference 
between  6  and  4  is  the  same  as  the  difference  between  10  and 
8.  And  the  numbers  6,  2,  12,  4,  are  in  geometrical  propor- 
tion, because  the  quotient  of  6  divided  by  2,  is  the  same  a^ 
the  quotient  of  12  divided  by  4. 

365.  Care  must  be  taken  not  to  confound  proportion  with 
ratio.  This  caution  is  the  more  necessary,  as  in  common 
discourse,  the  two  terms  are  used  indiscriminately,  or  rather, 

*  See  'Xote  L. 
24 


I8G  ALGEBRA. 

proportion  is  used  for  both.  The  expenses  of  oiie  man  are 
said  to  bear  a  greater  proportion  to  his  income,  than  those  of 
another.  But  according  to  the  defmition  which  has  just  been 
given,  one  proportion  is  neither  greater  nor  less  than  another. 
For  equality  does  not  admit  of  degrees.  One  ratio  may  be 
greater  or  less  than  another.  The  ratio  of  12  :  2  is  greater 
than  that  of  6  :  2,  and  less  than  that  of  20  :  2.  But  these  dif- 
ferences are  not  applicable  to  proportion,  when  the  term  is 
used  in  its  technical  sense.  The  loose  signification  which  is 
so  frecpiently  attached  to  this  word,  may  be  proper  enough  in 
familiar  language  :  for  it  is  sanctioned  by  a  general  usage. 
But  for  scientific  purposes,  the  distinction  between  proportion 
and  ratio  should  be  clearly  drawn,  and  cautiously  observed. 

366.  The  equality  betv/ecn  two  ratios,  as  has  been  stated, 
is  called  proportion.  Tlie  word  is  sometimes  applied  also  to 
the  series  of  terms  among  which  this  equality  of  ratios  exists. 
Thus  the  two  couplets  15:5  and  6  :  2  are,  when  taken  to- 
gether, called  a  proportion. 

367.  Proportion  may  ])e  expressed,  eitlier  by  the  common 
eign  of  equality,  or  by  four  points  between  the  two  couplets. 

rr,,  (  8  ••  6  =  4  ••  2,  or  8  ••  6  :  :  4  ••  2  ^  are  arithmetical 

Thus    ^ 


1 


a  ••  b  =  c  ••  d,  or  a--  b:  :  c  -•  d  )    proportions 


,      5  12:6  =  8:4,  or  12:  6:  :8:4  >  are  geometrical 
I    a:b  =  d:  h,  or    a:  b  ::  d:  h]    proportions. 

The  latter  is  read,  « the  ratio  of  a  to  b  equals  the  ratio  of  d 
to  A;'  or  more  concisely,  *  a  is  to  b,  as  d  to  /i.' 

363.  The  first  and  last  terms  are  called  the  extremes,  and 
the  other  two  the  means.  Homologous  terms  are  either  the 
two  antecedents  or  the  two  consequents.  Analogous  terms 
are  tlie  antecedent  and  consequent  of  the  same  couplet. 

369.  As  the  ratios  are  equal,  it  is  manifestly  immaterial 
which  of  the  two  couplets  is  placed  first. 

U  a  •  b  :  :  c  :  d,  then  c:  d:  :  a:b.     For  if  -—-  then  _=  -. 

b     d  do 

370.  The  number  of  terms  must  be,  at  least,  four.  For 
the  equality  is  between  the  ratios  of  tiro  couplets ;  and  each 
couplet  must  have  an  antecedent  and  a  consequent.  There 
may  be  a  proportion,  however,  among  three  quantities.     For 


PROPORTION.  137 

one  of  the  quantities  may  be  repeated,  so  as  to  form  two 
terms.  In  this  case  the  quantity  repeated  is  called  the  mid- 
dle tenn,  or  a  mean  proportional  between  the  two  other  quan- 
tities, especially  if  the  proportion  is  geometrical. 

Thus  the  numbers  8,  4,  2,  are  proportional.  That  is,  8  : 
4  :  :  4  :  2.  Here  4  is  both  the  consequent  in  the  first  couplet, 
and  the  antecedent  in  the  last.  It  is  therefore  a  mean  pro- 
portional between  8  and  2. 

The  last  term  is  called  a  third  proportional  to  the  two  other 
quantities.     Thus  2  is  a  third  proportional  to  8  and  4. 

371.  Inverse  or  reciprocal  \)VO\>OYtion  is  an  equality  l)etween 
a  direct  ratio,  and  a  reciprocal  ratio. 

Thus  4  :  2  : :  i  :  i  ;  that  is,  4  is  to  2,  reciprocally,  as  3  to  6. 
Sometimes  also,  the  order  of  tlie  terms  in  one  of  the  couplets, 
is  inverted,  without  writing  them  in  the  form  of  a  fraction. 
—(Art.  351.) 

Thus  4  :  2  :  :  3  :  6  inversely.  In  this  case,  the  first  term 
is  to  the  second,  as  the  fourth  to  the  third ;  tliat  is,  the  first 
divided  by  the  second,  is  equal  to  the  fourth  divided  by  the 
third. 

372.  When  there  is  a  series  of  quantities,  such  that  the 
ratios  of  the  first  to  the  second,  of  the  second  to  the  third,  of 
the  third  to  the  fourth,  &c.  are  all  equal;  the  quantities  are 
said  to  be  in  continued  proportion.  The  consequent  of  each 
preceding  ratio  is,  then,  the  antecedent  of  the  following 
one. — Continued  proportion  is  also  called  progression,  as  will 
be  seen  in  a  following  section. 

Thus  the  numbers  10,  8,  6,  4,  2,  are  in  continued  anthme- 
tical  proportion.     For  10  -  8:^8  -  6  =  6  -  4=4  -  2. 

The  numbers  64,  32,  16,  8,  4,  are  in  continued  geometncal 
proportion.     For  64  :  32  :  :  32  :  16  :  :  16  :  8  :  :  8  :  4. 

If  a,  6,  c,  d,  h,  &c.  are  in  continued  geometrical  propor- 
tion ;  then  a:  b  :  :  b  :c: :  c  :  d  :  :  d  :  h,  &:c. 

One  case  of  continued  proportion  is  that  of  three  propor- 
tional quantities.     (Art.  370.) 

373.  As  an  arithmetical  proportion  is,  generally,  nothing 
more  than  a  very  simple  equation,  it  is  scarcely  necessary  to 
give  the  subject  a  separate  consideration. 

The  proportion  a.  .b  :  :  c  .  .  d 

Is  the  same  as  the  equation         a-b=c-'d. 


188  ALGEBRA. 

It  will  be  proper,  however,  to  observe  that,  if  four  quanti- 
ties are  in  arithmetical  proportion,  the  sum  of  the  extremes  is^ 
equal  to  the  sum  of  the  means. 

Thus  if  a.  .b:  :  h.  .  m,  then  a-{-m=b-\-h 

For  by  supposition,  a-b  =  h-m 

And  transposing  -  6  and  -  m,  a-\-mz=b-\-h 

So  in  the  proportion,  12  . .  10 : :  1 1 . .  9,  we  have  12-f  9  =  10+1 1. 
Again  if  three  quantities  are  in  arithmetical  proportion,  the 
$um  of  the  extremes  is  equal  to  double  the  mean. 

If  a.  .  6  : :  i  . .  c,  then,  a-b  =  b-c 

And  transposing  -  6  and  -c,  a-\-c=2b. 

GEOMETRICAL  PROPORTION. 

374.  But  if  four  quantities  are  in  geometrical  proportion, 
the  PRODUCT  of  the  extremes  is  equal  to  the  product  of  the 
means. 

If  a  :  b  :  :  c  :  d,  ad=bc 

a_  c 
~b~d 
abd__cbd 

Reducing  the  fractions,  adz=bc 

Thus  12:8::  15:  10,  therefore  12x10=8x15. 
Cor.  Any  factor  may  be  transferred  from  one  mean  to  the 
other,  or  from  one  extreme  to  the  other,  without  affecting  the 
proportion.  If  a  :  mb  :  :x  :y,  then  a  :  b  :  :  mx  :  y.  For  the 
product  of  the  means  is,  in  both  cases  the  same.  And  if 
na  :  b  :  :  X :  y,  then  a:b:  :  x  :ny. 

375.  On  the  other  hand,  if  the  product  of  two  quantities 
is  equal  to  the  product  of  two  others,  the  four  quantities  will 
form  a  proportion,  when  they  are  so  arranged,  that  those  on 
one  side  of  the  equation  shall  constitute  the  means,  and  those 
on  the  other  side,  the  extremes. 

If  my=nh,  then  m:n::h  :  y,  that  is,  _=_ 

n     y 

For  by  dividing  my=nh  by  ny,  we  have       !?^zr:!L 


For  by  supposition,  (Arts.  346,  364.) 
Multiplying  by  bd,  (Ax.  3.) 


And  reducing  the  fractions, 


ny     ny 
h 

y 


PROPORTION.  189 

Cor.  The  same  must  be  true  of  any  factors  which  form  the 
two  sides  of  an  equation. 

If  {a-{-b)  Xc=:{d-m)  xy,  then  a-\-b  :  d  - m  :  :  y  :  c. 

376.  If  three  quantities  are  proportional,  the  product  of  the 
extremes  is  equal  to  the  square  of  the  mean.  For  this  mean 
proportional  is,  at  the  same  time,  the  consequent  of  the  first 
couplet,  and  the  antecedent  of  the  last.  (Art.  370.)  It  is 
therefore  to  be  multiplied  into  itself  that  is,  it  is  to  be  squared. 

If  a  :  b  :  :  b  :  c,  then  mult,  extremes  and  means,  ac=b'^. 

Hence,  a  mean  proportional  between  two  quantities  may  be 
found,  by  extracting  the  square  root  of  their  product. 

If  a  :  X  :  :  X  :  c,  then  x-=ac,  and  x=y\/ac.     (Art.  297.) 

377.  It  follows,  from  Art.  374,  that  in  a  proportion,  either 
extreme  is  equEil  to  the  product  of  the  means,  divided  by  the 
other  extreme  ;  and  either  of  the  means  is  equal  to  the  pro- 
duct of  the  extremes,  divided  by  the  other  mean. 


1. 

If  a  :  b  :  :c  :  ^,  then 

ad=hc 

2. 

Dividing  by  d, 

•4 

3. 

Dividing  the  first  by  c. 

c 

4. 

Dividing  it  by  b, 

"•i 

5. 

Divichng  it  by  a. 

i=t 

;     that    is,   the 
a 

fourth  term  is  equal  to  the  product  of  the  second  and  third 

divided  by  the  first. 

On  this  principle  is  founded  the  rule  of  simple  proportion 
in  arithmetic,  commonly  called  the  Rtde  of  Three.  Three 
numbers  are  given  to  find  a  fourth,  which  is  obtained  by 
multiplying  together  the  second  and  third,  and  dividing  by 
the  first. 

378.  The  propositions  respecting  the  products  of  the 
means,  and  of  the  extremes,  furnish  a  very  simple  and  con- 
venient criterion  for  determining  whether  any  four  quantities 
are  proportional.  We  have  only  to  multiply  the  means 
together,  and  also  the  extremes.  If  the  products  are  equal, 
the  quantities  are  proportional.  If  the  products  are  not  equal, 
the  quantities  arc  not  proportional. 


190  ALGEBRA. 

379.  In  mathematical  invest ig-ations,  when  the  relations 
of  several  quantities  are  given,  they  are  frequently  stated  in 
the  form  of  a  proportion.  But  it  is  commonly  necessary  that 
this  first  proportion  should  pass  through  a  n-uniber  of  trans- 
formations hefore  it  brings  out  distinctly  the  unknown  quan- 
tity, or  the  proposition  which  we  wish  to  demonstrate.  It 
may  undergo  any  change  wliich  will  not  affect  the  equality 
of  tJie  ratios  ;  or  whicli  will  leave  the  product  of  the  means 
equal  to  the  product  of  the  extremes. 

It  is  evident,  in  the  first  place,  that  any  alteration  in  the 
arrangement,  which  will  not  affect  the  equality  of  these  two 
products,  will  not  destroy  the  proportion.  Thus,  if  a  :  b  : :  c  :  d, 
the  order  of  these  four  quantities  may  be  varied,  in  any  way 
which  will  leave  ad— he.     Hence, 

380.  If  four  qnantities  are  proportional,  the  order  of 

THE  MEANS,  OR  OF  THE  EXTREMES,  OR  OF  THE  TERMS  OF 
BOTH  COUPLETS,  MAY  BE  INVERTED  AVITHOUT  DESTROYING 
THE  PROPORTION. 

If  a  :  b  :  :c  :  d  }  ,. 

And       12:S::6:4\^^'^''^ 

1.  Inverting  the  means,* 

a  :  c:  :b  :  d)    y       .^   (  The  first  is  to  the  third, 
1 2  :  6  :  :  S  :  4  5     ^^      ^'  ^  As  the  second  to  the  fourth. 
In  other  words,  the  ratio  of  the  antecedents  is  equal  to  the 
ratio  of  the  consequents. 

This  inversion  of  the  means  is  frequently  referred  to  by 
geometers,  under  the  name  of  Alternation. '\ 

2.  Inverting  the  extremes, 

d  :  b  :  :  c  :  a    >,        .     (  The  fourth  is  to  the  second, 
4  :  8 : :  6  :  12  i  ^^^""^  ^^'  (  As  the  third  to  the  first. 

3.  Inverting  the  terms  of  each  couplet, 

b  :  a    :  :  cZ  :  c  >    .       .     (  The  second  is  to  the  first, 
8  :  12  :  :  4  :  6  5  ^       ^^'  (  As  the  fourth  to  the  third. 
This  is  technically  called  Inversion. 

Each  of  these  may  also  be  varied,  by  changing  the  order 
of  the  two  couplets.     (Art.  369.) 

Cor.  The  order  of  the  ichole  proportion  may  be  inverted. 
If  a  :  b  :  :  c  :  d,  then  d  :  c :  :  b  :  a. 

*  See  Note  M.  f  Euclid,  16.  5. 


PROPORTION.  191 

In  each  of  these  case?,  it  will  be  at  once  seen  that,  by 
taking  the  products  of  the  means,  and  of  the  extremes,  we 
have  ad=bc,  and  12x4—8x6. 

If  the  terms  of  only  one  of  the  couplets  are  inverted,  the 
proportion  becomes  reciprocal     (Art  371.) 

If  a  :  b  :  :  c  :  d,  then  a  is  to  b,  reciprocally,  as  d  to  c. 
381.  A  difference  of  arrangement  is  not  the  only  alteration 
which  we  have  occasion  to  produce,  in  the  terms  of  a  pro- 
portion.   It  is  frequently  necessary  to  multiply,  divide,  involve, 
&c.     In  all  cases,  the  art  of  conducting  the  investigation 
consists  in  so  ordering  the  several  changes,  as  to  maintain  a 
constant  equality,  between  the  ratio  of  the  two  fiist  terms, 
and  that  of  the  two  last.     As  in  resolving  an  equation,  we 
must  see  that  the  sides  remain  equal ;  so  in  varying  a  pro- 
portion, the  equality  of  the  ratios  must  be  preserved.      And 
this  is  effected  either  bj^  keeping  the  ratios  the  same,  while 
the  terms  are  altered  ;  or  by  increasing  or  diminishing  one  of 
the  ratios  as  much  as  the  other.    Most  of  the  succeeding  proofs 
are  intended  to  bring  this  principle  distinctly  into  view,  and 
to  make  it  familiar.      Some  of  the  propositions  might  be  de- 
monstrated, in  a  more  simple  manner,  perhaps,  by  multiplying 
the  extremes  and  means.      But  this  would  not  give  so  clear 
a  view  of  the  nature  of  the  several  changes  in  tbe  proportions. 
It  has  been  shown  that,  if  both  the  terms  of  a  couplet  be 
multiplied  or  divided  by  the  same  quantit}",  the  ratio  will  re- 
main the  same  ;    (Art.  3G0.)  that  multiplying  the  antecedent 
is,  in  effect,  multiplying  the  ratio,  and  dividing  the  antece- 
dent, is  dividing  the  ratio  ;  (Art.  357.)  and  farther,  that  mul- 
tiplying the  consequent,  is,  in  effect,  dividing  the  ratio,  and 
dividing  the  consequent  is  multiplying  the  ratio.    (Art.  358.) 
As  the   ratios  in  a  proportion  are  equal,  if  they  are  both 
multiplied,  or  both  divided,  by  the  same  quantity,  they  will 
still  be  equal.    (Ax.  3.)   One  will  be  increased  or  diminished 
as  much  as  the  other.     Hence, 

382.  If  fom-  quantities  are  proportional,  two  analogous 
OR  two  homologous  terms  may  be  multiplied  or  di- 
vided BY  THE  SAME  QUANTITY,  WITHOUT  DESTROYING  THE 
PROPORTION. 

If  analogous  terms  be  multiplied  or  divided,  the  ratios  will 
not  be  altered.  (Art.  3G0.)  If  homologous  terms  be  multi- 
plied or  divided,  both  ratios  will  be  equally  increased  or 
diminished.      (Arts.  357,  8.) 


192  ALGEBRA. 

If  a  :  b  :  :  c  :  d,  then, 

1.  Multiplying  the  two  first  terms,  ma  :  mh  :  :  c  :  d 

2.  Multiplying  the  two  last  terms,  a  :  b:  :mc  :  md 

3.  Multiplying  the  tw^o  antecedents,*  ma  :  b  :  :  mc  :  d 

4.  Multiplying  the  two  consequents,  a  :  mb  : :  c  :  md 

5.  Dividing  the  two  first  terms,  _  :  _  :  :  c  :  ci 

m    m 

c      d 

6.  Dividmg  the  two  last  terais,  a  :  6  :  :  _  :  _ 

m     m 

7.  Dividing  the  two  antecedents,  _  :  6  :  :  _  :  rf 

m  m 

8.  Dividing  the  two  consequents,  a  '.  —  ::  c  :  — 

m  m 

Cor.  1.  All  the  terms  may  be  multiplied  or  divided  by  tbe 

same  quantity,  f 

,  .abed 

ma  :  mb  :  :mc  :  md^    _:_::_:  — 
w  mm     m     m 

\l^,  2.  In  any  of  the  cases  in  this  article,  multiplication 
of  tbe  consequent  may  be  substituted  for  division  of  the  ante- 
cedent ni  the  same  couplet,  and  division  of  the  consequent, 
for  multiplication  of  the  antecedent.      (Art.  359,  cor.) 

ma:b\'.mc\d\  |  \  a  :  —'.  :mc  :  d\       j  ma  :  6  :  :  c  :  _ 
Thus)  l"!]        rii  •      I       )  m 


_  :  6  :  :  —  :  ci     '^  \  a'.mb  :  :  —  :  d\       \  —  :b::c\  md 
m  m       J  ^  L  m      J       Lm 

383.  It  is  often  necessary  not  only  to  alter  the  terms  of  a 
proportion,  and  to  vary  the  arrangement,  but  to  compare  one 
proportion  with  another.  From  this  comparison  will  frequently 
arise  a  new  proportion,  which  may  be  requisite  in  solving  a 
problem,  or  in  carrying  forward  a  demonstration.  One  of 
the  most  important  cases  is  that  in  which  two  of  the  terms 
in  one  of  the  proportions  compared,  are  the  same  with  two  in 
the  other.  The  similar  terms  may  be  made  to  disappear, 
and  a  new  proportion  may  be  formed  of  the  four  remaining 
terms.     For, 


*  Euclid  3.  5.  t  Euclid  4.  5. 


PUOPORTION.  193 

384.  If  two  ratios  are  respectively  equal  to  a  third, 
they  are  equal  to  each  other.* 

This  is  nothing  more  than  the  11th  axiom  applied  to  ratios. 

1    If  a  :b::m:n}  ^^^en^.  ^  .  .  ^  .  j    ^.^ .  c::b:d.  (Art.  380.) 
And  c:a:  :m:n )  ^  ' 

2.  If      «:6::»':Mthena:6::c:rf,ora:c::6:i 
And  m  :  n  :  :  c  :  a  ) 

Cor.  K«:i^'»:»|thena:6>c:d.t 

For  if  the  ratio  of  m  :  n  is  greater  than  that  of  c:  dy  it  is 
manifest  that  tjtie  ratio  of  a  :  b,  which  is  equal  to  that  of  m  :  n, 
is  also  greater  than  that  of  c  :  c?. 

385.  In  these  instances,  the  terms  which  are  alike  in  the 
two  proportions  are  the  two  first  and  the  two  last.  But  this 
arrangement  is  not  essential.  The  order  of  the  terms  may- 
be changed,  in  various  ways,  without  affecting  the  equality 
of  the  ratios. 

1.  The  similar  terms  may  be  the  two  antecedents  or  the 
two  consequents f  in  each  proportion.     Thus, 

If      m:  a:  :n:b  }   -,        (By  alternation,  m  :  n  .  .  ^:b 
And  m\  c:  :  h:  d)  \  And  m  :  :<i :  :  c  :  d 

Therefore  a  :  6  : :  c  :  c?,  or  a  :  c  :  :  6  :  d,  by  the  last  article. 

2.  The  antecedents  in  one  of  the  proportions,  ma)n)e  the 
same  as  the  consequents  in  the  other. 

If      m  :  a  :  :  n :  6  >    ,        (By  inver.  and  altern.  a:b::m:n 
And  c:m::d:n)  ^  By  alternation,  c:  d::m:n 

Therefore  a  :  b,  &c.  as  before. 

3.  Two  homologous  terms,  in  one  of  the  iproportions,  may- 
be the  same,  as  two  analogous  terms  in  the  Aher. 

If      a  :  m :  :  6  :  n  >   ,         (By  aUernation,  a  :b:  :m:n 
And  c:d::m:nS  (  And  c:  d::m:n 

Therefore,  a  :  6,  &c . 

All  these  are  instances  of  an  equality,  between  the  ratios  in 
one  proportion,  and  those  in  another.     In   geometry,   the 


♦  Euclid  n.  5.  t  Euclid  13.  5. 

25 


^ 


194  ALGEBRA. 

proposition  to  which  they  belong  is  usually  cited  by  the 
words  "ea;  aequo,^^  or  "ea?  aequali.""^  The  second  case  in 
this  article  is  that  which  in  its  form,  most  obviously  answers 
to  the  explanation  in  Euclid.  But  they  are  all  upon  the 
same  principle,  and  are  frequently  referred  to,  without  dis- 
crimination. 

386.  Any  number  of  proportions  may  be  compared,  in  the 
same  manner,  if  the  two  first  or  the  two  last  terms  in  each 
preceding  proportion,  are  the  same  with  the  two  first  or  the 
two  last  in  the  following  one.* 

Thus  if  a  :  b  :  :  c  :  d'] 

And        c :  d:  :h:l  [  ,^  i 

KALI  }  then  a  :  b  :  :  X :  y.  • 

And        h:  l::m:n  (  ^ 

And       mm:  :  x  :y  j 

That  is,  the  two  first  terms  of  the  first  proportion  have  the 

same  ratio,  as  the  tw^o  last  terms  of  the  last  proportion.     For 

it  is  manifest  that  the  ratio  of  all  the  couplets  is  the  same. 

And  if  the  terms  do  not  stand  in  the  same  order  as  here, 
yet  if  they  can  be  reduced  to  this  form,  the  same  principle  is 
applicable. 

Thusif  a:c:  :6:  c?]  ~       faibiicid 

And  c:h::d:l\  ^^      ^    alternation  \  f  ;  f  .V  j,^ !  [ 

And  him::  l:n  (  ^  ^  h:  l::m:n 

Ajid  m:  X  :  :n:y  ]  im:n::x:y 

'^*  Therefore  a:b::  x:y,  as  before. 

In  all  the  examples  in  this,  and  the  preceding  articles,  the 
two  terms  in  one  proportion  which  have  equals  in  another, 
are  neither  the  two  means,  nor  the  two  extremes,  but  one  of 
the  means,  and  one  of  the  extremes ;  and  the  resulting  pro- 
portion is  umfoi'miy^  direct. 

387.  But  if  the  two  means,  or  the  two  extremes,  in  one 
proportion,  be  the  same  with  the  means,  or  the  extremes,  in 
another,^  the  four  remaining  terms  will  be  reciprocally  propor- 
tional. 

If      a:m:  :  n:  b  )  ,, 1.1 

And  c:m 

For     ab= 


»    J  J  >  then  a:c::-  :  -.  or  a  :  c  :  :  d  :  b. 

And  c:m:  :n:  d  )  b     d 

For     ab=mn  >  .^^^^  3^^  .  Therefore  a,b  =  cd,  Rnda:c::d:b. 
And    cd=mn  S 


*  Euclid  22.  6. 


PROPORTION.  VJ5 

In  this  example,  the  two  means  in  one  proportion,  are  hke 
those  in  the  other.  But  the  principle  will  he  the  same,  if  the 
extremes  are  alike,  or  if  the  extremes  in  one  proportion  are 
like  the  means  in  the  other. 


If      m  :  a  :  :  b  .  ,„  r    ^,  ,    , 

then  a  :  c  :  :  a  :  0. 


:  a::  b  :n) 
And  m:  c:  :  dm  ) 


Or  if  a  :  m  :  :  n  :  b  }  ^.  ,    , 

*     1  J       }  then  a  :  c  :  :  a  :  b. 

And  m:  c  : : a:n  ) 

The  proposition  in  geometry  which  applies  to  this  case,  is 
usually  cited  by  the  words  "  ex  aequo  perturbatey* 

388.  Another  way  in  which  the  terms  of  a  proportion  may 
be  varied,  is  by  addition  or  subtraction. 

If  to  or  from  two  anal'ogous  or  two  homologous 
terms  of  a  proportion,  two  other  quantities  having 
the  same  ratio  be  added  or  subtracted,  the  proportion 

WILL  BE  PRESERVED. f 

For  a  ratio  is  not  altered,  by  adding-  to  it,  or  subtracting 
from  it,  the  terms  of  another  equal  ratio.     (Art.  362.) 

If      a:b:  :  c  :  d 
And  a:  b::m:n 

Then  by  adding  ^  or  subtracting  from  a  and  6,  the  terms 
of  the  equal  ratio  m.n,  \ve  have, 

a-|-m  :  b-{-n :  :  c:  d,  and  a-mib-niicd. 

And  by  adding  and  subtracting  m  and  n,  to  and  frofti'c  and 
d  we  have, 

a:  b  :  :  c-j-wi :  d-\-n,  and  a  :  b  :  :c-m:  d-n. 

Here  the  addition  and  subtraction  are  to  and  from  analo- 
gous terms.  But  by  alternation,  (Art.  380,)  these  terms  will 
become  homologous,  and  w^e  shall  have, 

a-{-m  :  c  :  :  b-\-n  :  d,  and  a-  m  :  c  :  :  b—n:  d. 

Cor.  1.  This  addition  may,  evidently,  be  extended  to  any 
number  of  equal  ratios.  X 

rc:d  ,       ^ 

Thusifa:6::<  ^*  ^ 

[x:y 
Then  a :  6  :  :  c-{-h-{-m-\-x  :  d-\-l-\-n-\-y. 

*  Euclid  23,  5.  t  Euclid  2,  5.  t  Euclid  2,  5.    Cor. 


196  ALGEBRA. 

Cor.  2.  \{a:b::c:d},  ,        ,         ,        ,^ 

41        ,  ,  >  then  a-\-m  :  0  :  :  c-\-n :  a.^ 

And  7»  :  6  :  :  71 :  a  5 

For  by  alternation  a:  c  : :  b:  d  )  there-  (      a-\-m  :  c-^n  ::b:  d 
And  m:  n  :  :  b  :  d)    fore    (  or  a-f-w :  6  : :  c+w :  ^. 

389.  From  the  last  article  it  is  evident  that  if,  in  any  pro- 
portion, the  terms  be  added  to,  or  subtracted  from  each  otheVy 
that  is. 

If  two  analogous  or  homologous  terms  be  added  to, 
or  subtracted  from  the  two  others,  the  proportion 
will  be  preserved. 

Thus,  if  a:  b::c:d,  B,nd  12:4:  :6:2,  then, 

1.  Adding  the  two  last  terms,  to  the  two  first. 

a-\-c:b-\-d::a\b  I24-6  :    4+2::  12:  4 

and  a-f-c  :  6-l-d  :  :  c  :  cl  12+6:    4  +  2::    6:2 

or  a+c:  a::  6+d:  6  12+6:  12::  4+   2:4 

and  a+c:c::6+c^  :  d  12+6:    6::  4+   2:2. 

2.  Adding  the  two  antecedents,  to  the  two  consequents. 

a-\.b:b:'.c-\-d'.d  12+4:    4  :  :  6+2  :  2 

a+6  :  a  :  :  c+rf  :  c,  &c.       12+4 :  12  :  :  6+2  :  6,  &c. 
This  is  called  Composition.\ 

3.  Subtracting  the  two  first  terms,  from  the  two  last. 

c-a: a :  :  d- b : b 
c-a:  c  :  :  d-b  :  d,  &c. 

4.  Subtracting  the  two  last  terms  from  the  two  first. 

a-c  :  b-d:  :a:  bi 
a-c:b-d'.  :  c  :  d,  &c. 

5.  Subtracting  the  consequents  from  the  antecedents. 

a-b  :  b  :  :  c-d:  d 
a  :  a  -  b  :  :  c  :  c  -  d,  &c. 
The  alteration  expressed  by  the  last  of  these  forms  is  called 
Conversion. 

6.  Subtracting  the  antecedents  from  the  consequents. 

b-a  :  a: :  d  -  c:  c 

b  :  b-a: :  d  :  d-c,  &c. 

♦  Euclid  24,  5.  f  Euclid  18,  5.  J  Euclid  19,  6. 


PROPORTION.  197 

7.  Adding  and  subtracting, 

a-\-b  :  a-b  :  :  c-\-d  :  c-d. 

That  is,  the  sum  of  the  two  first  terms,  is  to  their  differ- 
ence, as  the  smn  of  the  two  last,  to  their  difference. 

Cor.  If  any  compound  quantities,  arranged  as  in  the  prece- 
ding examples,  are  proportional,  the  simple  quantities  of  which 
they  are  compounded  are  proportional  also. 

Thus,  if  a-\-b  :  b  ::  c-\-d :  d,  then  a  :  b  :  :  c  :  d. 

This  is  called  Division.'^ 

390.  If  the  corresponding  terms  of  two  or  more 

RANKS     OF     proportional     QUANTITIES    BE    MULTIPLIED 
TOGETHER,  THE    PRODUCT     WILL    BE    PROPORTIONAL. 

This  is  compounding  ratios,  (Art.  352,)  or  compounding 
proportions.  It  should  be  distinguished  from  v,  hat  is  called 
composition^  which  is  an  addition  of  the  terms  of  a  ratio.  (Art. 
389.  2.) 

If         a:b::c:dl  12  :  4:  :6  :  2  > 

And     h:l::m:nl  10:5::8:45 


Tlien  ahibl::  cm  :  dn  120  :  20 :  :  48  :  8. 

For  from  the  nature  of  proportion,  the  two  ratios  in  the 
first  rank  are  equal,  and  also  the  ratios  in  the  second  rank. 
And  multiplying  the  corresponding  terms  is  multiplying  the 
raiioSf  (Art.  357.  cor.)  that  is,  multipl}ing  equcds  by  equals  ; 
(Ax.  3.)  so  that  the  ratios  will  still  be  equal,  and  therefore 
the  four  products  must  be  proportional. 

The  same  proof  is  applicable  to  any  number  of  proportions. 

Ca:b::c:d 
If  }h:l::m:n 

ip:q::x:y 
Then  ahp  :  blq  :  :  cmx  :  dny. 

From  this  it  is  evident,  that  if  the  terms  of  a  proportion  be 
multiplied,  each  into  itself,  that  is,  if  they  be  raised  to  any 
poxcer,  they  will  still  be  proportional. 

U  a:  b::c  :  d  2 :  4::6  :  12 

a:b::c:  d  2  :4::6  :  12 


Then  a"  :  b""  :  :  c^  :  d'  4  :  16  : :  36  :  144 


*  Euclid  17.  5.     See  Note  N. 


198  ALGEBRA. 

Proportionals  will  also  be  obtained,  by  reversing  this  pro- 
cess, that  is,  by  extracting  the  roots  of  the  terms. 

U  a  :  h  :  :  c  :  d,  then  \/a  :  \/b  :  :  \/c  :  \/d. 

For  taking  the  product  of  extr.  and  means,       ad=hc 
And  extracting  both  sides,  \fad-=A^bc 

That  is,  (Arts.  259,  375.)  .^a  :  ^/b  : :  a^c  :  A^d. 

Hence, 

391.  If  several  quantities  are  proportional,  their  like 

POWERS    OR    LIKE    ROOTS    ARE    PROPORTIONAL.* 

\i  a:  b  :\c:  d 
Then  «"  :  6" :  :  c"  :  d\  and  li^a  :\/b  ::yc  :  "Jijd. 

mm  mm 

And  V«"  :  V^"  •  •  a/^"  •  V^^  ^^^^^  i^*  '^  :b^::e  id'*. 

392.  If  the  terms  in  one  rank  of  proportionals  be  divided 
by  the  corresponding  terms  in  another  rank,  the  quotients 
will  be  proportional. 

This  is  sometimes  called  the  resolution  of  ratios. 

If      a:b::c:dl  12  :  6::  18  :  9  ) 

And  h:l::m:nS  6:2::    9:35 

Then  ^-^  "  1  •-  l^-?--!?-? 

^^h'  I  "  m'  n  6   '  2 '  *  9   *  3 

This  is  merely  reversing  the  process  in  Art.  390,  and  may 
be  demonstrated  in  a  similar  manner. 

This  should  be  distinguished  from  what  geometers  call 
division,  which  is  a  subtraction  of  the  terms  of  a  ratio.  (Art. 
389.  cor.) 

When  proportions  are  compounded  by  multiplication,  it 
will  often  be  the  case,  that  the  same  factor  will  be  found  in 
two  analogous  or  two  homologous  terms. 
Thus  if  a:  b::c:  d) 
And       m  :  a : :n:  c  3 


am  :  ab  wen  :  cd. 


Here  a  is  in  the  two  first  terms,  and  c  in  the  two  last.    Di- 
viding by  these,  (Art.  382,)  the  proportion  becomes 
m  :  b  :  :n  :  d.     Hence, 


*  It  must  not  be  inferred  from  this,  that  quantities  have  the  same  ratio  as 
their  like  powers  or  like  roots.    See  Art.  354. 


PROPORTION.  199 

393.  In  compounding  proportions,  equal  factors  or  divisors 
in  two  analogous  or  homologous  terms,  may  be  rejected. 

Ca:  b::c:  d  12  :  4: :9  :  3 

If  }b  :h::d:  I  4  :  8  :  :  3  :  6 

(h:m::l:n  8:20::6  :15 


Thena:m::c:n  12  :  20::9  :  15 

This  rule  may  be  applied  to  the  cases,  to  which  the  terms 
"  ex  aequo''  and  "  ex  aequo  perturb  ate'"'  refer.  See  Arts.  385  and 
387.     One  of  the  methods  may  serve  to  verify  the  other. 

394.  The  changes  which  may  be  made  in  proportions, 
without  disturbing  the  equality  of  the  ratios,  are  so  nmiie- 
rous,  that  they  would  become  burdensome  to  the  memory,  if 
they  were  not  reducible  to  a  few  general  principles.  They 
are  mostly  produced, 

1 .  By  inverting  the  order  of  the  terms,  Art.  380. 

2.  By  multiplying  or  dividing  by  the  same  quantity.  Art.  382. 

3.  By  comparing  proportions  which  have  like  terms.  Art.  384, 

5,  6,  7. 

4.  By  adding  or  subtracting  tlie  terms  of  equal  ratios,  Art. 

388,  9. 

5.  By  multiplying  or  dividing  one  proportion  by  another,  Art. 

390,  2,  3. 

6.  By  involving  or  extracting  the  roots  of  the  terms.  Art.  391. 

395.  When  four  quantities  are  proportional,  if  the  first  be 
greater  than  the  second,  the  third  will  be  greater  than  the 
fourth ;  if  equal,  equal :  if  less,  less. 

For,  the  ratios  of  the  two  couplets  being  the  same,  if  one  is 
a  ratio  of  equality,  the  other  is  also,  and  therefore  the  ante- 
cedent in  each  is  equal  to  its  consequent ;  (Art.  350,)  if  one 
IS  a  ratio  of  greater  inequality,  the  other  is  also,  and  therefore 
the  antecedent  in  each  is  greater  than  its  consequent ;  and 
if  one  is  a  ratio  of  lesser  inequality,  the  other  is  also,  and 
therefore  the  antecedent  in  each  is  less  than  its  consequent. 

(  a—b,  c  =  d 

Let  a:b  :  :  c  :  d;  then  if  <  a^b,  c^d 

f  a<^b,  c<^d. 


200  ALGEBRA. 

Cor.   1.   If  the  first  be  greater  than  the  thirds  the  second 
will  be  greater  than  the  fourth ;  if  equal,  equal ;  if  less,  less.* 

For  by  alternation,  a  :  h  ::c  :  d  becomes  a  :  c'.:h  :  d,  with- 
out any  alteration  of  the  quantities.  Therefore,  if  a=6, 
c=di  &c.  as  before. 

Cor.  2.  If    rt  :  m  :  :  c  :  n  >  ,1       .^         ,  j   o     j. 

,         ,  ,\  then  if  a=h.  c  =  d,  &c.t 

and  m :  0 : :n:  a )  '  '         ' 

For,  by  equality  of  ratios,  (Art.  385.  2.)  or  compounding 
ratios,  (Arts.  390,  393.) 

a  :  b  : :  c  :  d.     Therefore,  if  a=zb,c=d,  &c.  as  before. 

Cor.  S.  If    a  :  m  :  :n  :  d  )  .■,        -r        j  i    o     x 

,         ,  >  then  if  a=b,  c=d,  &c.i 

and  m  :  0'.  :  c  \  n)  ^ 

For,  by  compounding  ratios,  (Arts.  390,  393,) 

a  :  b::c  :  d.     Therefore,  if  a=b,  c=d,  &c. 

395.  b.  If  four  quantities  are  proportional,  their  reciprocals 
are  proportional  ;  and  v.  v. 

li  a  :b  : :  c  :  d,  then  _:_::_:_. 
a     b      c     d 

For  in  each  of  these  proportions,  we  have,  by  reduction, 
ad=zbc.  ^ 


CONTINUED  PROPORTION. 

396.  When  quantities  are  in  continued  proportion,  all  the 
ratios  are  equal.     (Art.  372.)     If 

a  :  b  :  :b  :  c::c  :  d:  :  d  :  e, 
the  ratio  of  a  :  6  is  the  same,  as  tliat  of  b  :  c,  of  c  :  d,  or  of 
d  :  e.     The  ratio  of  the  first  of  these  quantities  to  the  last,  is 
equal  to  the  product  of  all  the  intervening  ratios  ;   (Art.  353,) 
that  is,  the  ratio  of  a  :  e  is  equal  to 

b     c     a     e 
But  as  the  intervening  ratios  are  all  equal,  instead  of  multi- 
plying them  into  each  other,  we  may  multiply  any  one  of 
them  into  i^se//' ;  observing  to  make  the  number  of  factors 


♦  Euclid  14.  5.  t  Euclid  20.  5.  t  Euclid  21.  5. 


PROPORTION.  201 

equal  to  the  number  of  intervening  ratios.      Thus  the  raiio 
of  a  :  e,  in  the  example  just  given,  is  equal  to 

When  several  quantities  are  in  continued  proportion,  the 
number  of  couplets,  and  of  course  the  number  of  ratios,  is 
one  less  thaj/the  number  of  quantities.  Thus  the  five  pro- 
porti^p^Jiantities  a,  6,  c,  d,  e,  form  four  couplets  containing 
four  ^nos  ;  and  the  ratio  of  a  :  e  is  equal  to  the  ratio  of 
a*  :  b\  that  is,  the  ratio  of  the  fomth  power  of  tlie  fiist  quan- 
tity, to  the  fourth  poAver  of  the  second.     Hence, 

397.  If  three  quantities  are  proportional,  the  first  is  to  tlie 
third,  as  the  square  of  the  first,  to  the  square  of  the  second ;  or 
as  the  square  of  the  second,  to  the  square  of  the  third.  In 
other  words,  the  first  has  to  the  third,  a  duplicate  ratio  of  the 
first  to  the  second.  And  conversely,  if  the  first  of  the  three 
quantities  is  to  the  third,  as  the  square  of  the  first  to  the 
square  of  the  second,  the  three  quantities  are  proportional. 

If  a  :  6  :  :  6  :  c,  then  a  \  c:  \  a?  :  h"^.     Universally, 

398.  If  several  quantities  are  in  continued  proportion,  the 
ratio  of  the  first  to  the  last  is  equal  to  one  of  the  intervening 
ratios  raised  to  a  power  whose  index  is  one  less  than  the  nmn- 
ber  of  quantities. 

If  there  are  four  proportionals  a,  b,  c,  d,  then  a  :  d  :  :  d^  :  b^ 
If  there  are  five  a,  b,  c,  d,  e ;  a  :  e  :  :  a^  :  b*,  Slc. 

399.  If  several  quantities  are  in  continued  proportion,  they 
will  be  proportional  when  the  order  of  tlie  whole  is  inve^rted. 
This  has  already  been  proved  with  respect  to  four  proportional 
quantities.  (Art.  380.  cor.)  It  may  be  extended  to  any  num- 
ber of  quantities. 


Between  the  numbers. 

64,33,  16,8,4, 

The  ratios  are 

2,2,2,2, 

Between  the  same  inverted 

4,8,  16,32,  64, 

The  ratios  are 

1111 

2>    2'    2J    2' 

So  if  the  order  of  any  proportional  quantities  be  inverted,' 
the  ratios  in  one  series  will  be  the  reciprocals  of  those  in  the 
other.  For  by  the  inversion,  each  antecedent  becomes  a  con- 
sequent, and  V.  V.  and  the  ratio  of  a  consequent  to  its  antece- 
dent is  the  reciprocal  of  the  ratio  of  the  antecedent  to  the 

26 


202  ALGEBRA. 

consequent.  (Art.  351.)  That  the  reciprocals  of  equal  quan- 
tities are  themselves  equal,  is  evident  from  Ax.  4. 

400.  Harmonical  or  Musical  Proportion  may  he  con- 
sidered as  a  species  of  geometrical  proportion,  it  consists  in 
an  c(iuality  of  geometrical  ratios  ;  hut  one  or  more  of  the 
terms  is  tlie  difference  between  two  quantities. 

TJiree  or  four  quantities  are  said  to  he  m  harmonical  propor^ 
tion,  when  the  first  is  to  the  last,  as  the  difierence  hetween 
the  tivo  first,  to  the  dilfeience  hetween  the  two  last. 

If  tlie  three  quantities  a,  b,  and  c,  are  in  harmonical  pro- 
portion, then  a  :  c  :  :  a-b  :  b-c. 

If  the  four  quantities  a,  b,  c,  and(?,  are  in  harmonical  pro- 
portion, tlien  a  :  d  :  :  a-b  :  c-d. 

Thus  the  three  numbers  12,  8,  6,  are  in  harmonical  pro- 
portion. 

And  the  four  numbers  20.  16,  12,  10,  are  in  harmonical 
proportion. 

401.  If,  of  four  quantities  in  harmonical  proportion,  any 
three  be  given,  the  other  may  be  found.  For  from  the  pro- 
portion, 

a  :  d  : :  a-b  :  c  -d, 

by  taking  the  product  of  the  extremes  and  the  means,  we 
have  ac-ad  —  ad-hd. 

And  this  equation  may  be  reduced,  so  as  to  give  the  value 
of  either  of  the  four  letters. 

Thus  by  transposing  -  ad,  and  dividing  by  a, 

'2,ad  -  bd 
c  = 

a 

Examples,  in  which  the  principles  of  proportion  are  applied  to  the 
solution  of  problems. 

1.  Divide  the  number  49  into  two  such  parts,  that  the 
greater  increased  by  6,  may  be  to  the  less  diminished  by  1 1  ; 
as  9  to  2. 

Let  x=  the  greater,         and  49  -a:  =  the  less. 
By  the  conditions  proposed,  x-\-6  :  38-a::  :  9  :  2 

Adding  ternis,  (Art.  389,  2.)  x-^Q  :  44  :  :  9  :  11 

Dividing  the  consequents,  (Art.  382,  8.)  x-\-6  :  4  : :  9  :  1 
Multiplying  the  extremes  and  means,  a:+6  =  36.  And  a?=30. 


PROPORTION.  203 

2.  What  number  is  that,  to  which  if  1,  5,  nnd  13,  be  seve- 
rally added,  the  fiust  sum  shall  be  to  the  second,  as  the  sec- 
ond to  the  tliird  '^ 

Let  x=  the  number  required. 

By  the  conditions,  a:-|-l  :  .t4-5  : :  x-{-5  :  rr+lS 

Subtracting  terms,  (Art.  389,  6.)  x-\-l  :  4  :  :  x-\-5  :  8 
Therefore  82^4-8= 4x4-20.    And  a: =3. 

3.  Find  two  numbers,  the  gfi-eater  of  which  shall  be  to  the 
less,  as  their  sum  to  42  ;  and  as  their  diiference  to  6. 

Let  X  and  yz=  the  numbers. 

42 
6 
6 
6 

36 
3 


By  the  conditions,  x  :  y  ::  x-\-y 

And  X  :y  :  :  x-y 

By  equality  of  ratios,  x~\-y  :  42  :  :  x-y 

Inverting  the  means,  x-\-y  :  x-y  :  :42 

Adding  and  subtracting  terms, (Art.  389,  7,)  2x  :2y  ::4S 
Dividing  terms,  (Art.  382,)  x  :y::4 

Therefore  3x=4y.     And  x=^ 

^  3 

From  the  second  proportion,  Qx=zyx{x-y) 

Substituting  -^  for  x,  yz=24.     And  a: =32. 

o 

4.  Divide  the  number  18  into  two  such  parts,  tliat  the 
squares  of  those  parts  may  be  in  the  ratio  of  25  to  IG. 

Let  x=  the  gi'eater  part,  and  18  -  a:=:  the  less. 
By  the  conditions,  x""  :  {IS- xY:  :2o  :  16 

Extracting,  (Art.  391,)  a;  :  18-a:: :  5  :    4 

Adding  terms,  a; :  18  :  :  5  :    9 

Dividing  terms,  x:    2::  5:    1 

Therefore,  a:  =10. 

5.  Divide  the  number  14  into  two  such  parts,  that  the  quo- 
tient of  the  greater  divided  by  the  less,  shall  be  to  the  quotient 
of  the  less  divided  by  the  greater,  as  i6  to  9. 

Let  x=z  the  greater  part,  and  14-a:=  the  les*. 


204 

ALGEBRA. 

By  the  conditions, 

X 

X             X 

:  16  :  9 

Multiplying  terms, 
Extracting, 

x' 
X  ; 

'.{U-xy: 
14-a;::4  : 

:16:9 
3 

Adding  terms, 
Dividing  terms. 
Therefore, 

x  : 
X  : 

14::4:  7 
2::4:  1 

a;=8. 

6.  If  the  number  20  be  divided  into  two  parts,  which 
are  to  each  other  in  the  duplicate  ratio  of  3  to  1,  what  num- 
ber is  a  mean  proportional  between  those  parts  1 

Let  x=  the  greater  part,  and  20-  a:  =  the  less. 

By  the  conditions,  a;  :  20  -  or : :  3' :  P  : :  9  :  6 

Adding  terms,  a;  :  20  :  :  9  :  10 

Therefore,  a;=:18.     And20-a;=2 

A  mean  propor.  between  18  and  2  (Art.  376.)  =  /y/2><T8=6. 

7.  There  are  two  numbers  whose  product  is  24,  and  the 
difference  of  their  cubes,  is  to  the  cube  of  their  difference,  as 
19  to  1.     What  are  the  numbers  ] 

Let  x  and  y  be  equal  to  the  two  numbers. 

1.  By  supposition,  xy=M\ 

2.  And  x^-'if  :  {x-yf:  :  19  :  1  5 

3.  Or,  (Art.  217.)  x^-"^'  x^-Sx'y-^Sxy^-y^: :  19:1 

4.  Therefore,  (Art.  389,  5,)        Sx'y-Sxy''  :  (x-yY::  18  ,  I 

5.  Dividing  by  x-y  (Art.  382,  5,)        Sxy:  {x-yyiilS:  1 

6.  Or,  as  Sxy  =  Sx^^=72,  72  :  {x-yY  : :  18  :  1 

7.  Multiplying  extremes  and  means,  {x-yY=4 

8.  Extracting,  x-y=  2> 

9.  By  the  first  condition,  we  have  xy=24l 
Reducing  these  two  equations,  we  have     x=6,  and  i/=4. 

8.  It  is  required  to  prove  that        a  :  x  ::  ^y2a-y  :  r^y 
on  supposition  that  («+a?)^  :  (a  -  a;)' :  :  x-\~y  :  x-y. 


^ 


♦Bridge's  Algebra, 


PROPORTION.  205 

1.  Expandirig,           a'-\-2ax-\-x'^ :  a^  -  2ax-\-x^ :  :  x-]-y  :  x  -y 

2.  Adding  and  subtracting  terms,  2a^-\-2x^  :  4ax  :  :2x:  2y 

3.  Dividing  terms,  a^-j-.r'^ :  2ax :  :  x  :y 

4.  Transf.  the  factor  x,   (Art.  374.  cor.)     a^-\-x'  :2a::x''  :y 

5.  Inverting  the  means,  a^-\-x^  :ar^:  :2a:y 

6.  Subtracting  terms,  d^ :  x"^ : :  2a  -  y  :  y 

7.  Extracting,  a:x  :  :  ^2a-y  :  ^y 

9.  It  is  required  to  prove  that  dx=cy,  if  x  is  to  y  in  the 
tripUcate  ratio  of  a  :  6,  and  a:b  :  :  \/c-\-x  :  :  \/d-\-y. 

1.  Involving  terms,  a^  :b^ :  :  c-\-x  :  (l-{-y 

2.  By  the  first  supposition,  d^ :  b^ :  :  x:y 

3.  By  equality  of  ratios,  c-\-x  :  d-{-y  :  :  x  :y 

4.  Inverting  the  means,  c-\-c§  :  x  :  :  d-\-y  :  y 

5.  Subtracting  terms,  c  :  x:  :  d:y 

6.  Therefore,  dx=cy. 

10.  There  are  two  numbers  whose  product  is  135,  and  the 
difference  of  their  squares,  is  to  the  square  of  their  difference, 
as  4  to  1.     What  are  the  numbers  1  Ans.  15  and  9. 

1 1 .  What  two  numbers  are  those,  whose  difference,  sum, 
and  product,  are  as  the  numbers  2,  3,  and  5,  respectively  1 

Ans.  10  and  2. 

12.  Divide  the  number  24  into  two  such  parts,  that  their 
product  shall  be  to  the  sum  of  their  squares,  as  3  to  10. 

Ans.  18  and  6. 

13.  In  a  mixture  of  rum  and  orandy,  the  difference  be- 
tween the  quantities  of  each,  is  to  the  quantity  of  brandy,  as 
100  is  to  the  number  of  gallons  of  rum ;  and  the  same  dif- 
ference is  to  the  quantity  of  rum,  as  4  to  the  number  of 
gallons  of  brandy.     How  many  gallons  are  there  of  each  1 

Ans.  25  of  rum,  and  5  of  brandy. 

14.  There  are  two  numbers  which  are  to  each  other  as  3 
to  2.  If  6  be  added  to  the  greater  and  subtracted  hom  the 
less,  the  sum  and  remainder  will  be  to  each  other,  as  3  to  1. 
What  are  the  numbers  ?  Ans.  24  and  16. 

15.  There  are  two  numbers  whose  product  is  320  ;  and  the 
difference  of  their  cubes,  is  to  the  cube  of  their  difference,  as 
61  to  1.     What  are  the  numbers]  Ans.  20  and  16. 


206  ALGEBRA. 

16.  Tliere  are  two  immbeis,  which  are  to  each  other,  in 
tlie  duplicate  ratio  of  4  to  3  ;  and  24  is  a  mean  proportional 
between  them.  AVhat  are  the  numbers  ?    Ans.  32  and  18. 

402.  A  hst  of  the  articles  in  this  section  which  contain  the 
projwsitions  in  the  5th  book  of  EucUd.* 

Prop.  I.         Art.  363.  XIII.  384,  cor. 

II.  388.  XIV.  395,  cor.  1. 

III.  382.  XV.  360. 

IV.  382,  cor.  1.  XVI.  380. 

V.  382.     ,  XVII.  389,  cor. 

VI.  362.  XVIII.  ,89,  2. 

VII.  349,  cor.  1.  XIX.  389,4. 
VIIL  357,  cor.  358,  cor.  XX.  395,  cor.  2. 

IX.  349,  cor.  2.  XXI.  395,  cor.  3. 

X.  357,  cor.  358,  cor.  XXII.         386. 

XI.  384.  XXIII.       387. 

XII  363.  XXIV.        388.  cor.  2. 


SECTION  XIII. 


VARIATION  OR  GENERAL  PROPORTION.! 

Art.  403.  THE  quantities  which  constitute  the  tenns  of 
a  proportion  are,  frequently,  so  related  to  each  other,  that,  if 
one  of  them  be  either  increased  or  dhninished,  another  de- 
pending- on  it  will  also  be  increased  or  diminished,  in  such  a 
manner,  that  the  proportion  will  still  be  preserved.  If  the 
value  of  50  yards  of  cloth  is  100  dollars,  and  the  quantity 
be  reduced  to  40  yards ;  the  value  will,  of  coiuse,  be  reduced 
to  80  dollars ;  if  the  quantity  be  reduced  to  30  yards,  the 
value  will  be  reduced  to  60  dollars,  &c. 


♦  See  note  O. 

t  Newton's  Princip.  Book  I.  Sec.  I.  Lemma  10,  schol.  Emerson  on  Pro- 
portion, Wood's  Algebra,  Ludlam's  Math.,  Saunderson's  Algebra,  Art,  290, 
^rkinson's  Mechanics,  p,  24. 


VARIATION.  207 

ya.   yd.     dol.    dot 
That  is,  50:40::  100:80 
50:30:  :  100:60 
50:20:  :  100:40,  &c. 
As  the  consequent  of  the  first  couplet  is  varied,  the  conse- 
quent of  ihe  second  is  varied,  in  such  a  manner,  that  tlie  pro- 
portion is  constantly  preserved. 

If  the  two  antecedents  are  A  and  B ;  and  if  a  represents  a 
quantity  of  the  same  kind  with  A,  hut  eitjier  greater  or  less ; 
and  b,  a  quantity  of  the  same  kind  with  B,  but  as  many  times 
greater  or  less,  as  a  is  greater  or  less  than  A ;  then 

A:a::B:b; 
that  is,  if  A  by  varying  becomes  a,  then  B  becomes  b.  This 
is  expressed  more  concisely,  by  saying  that  A  varies  as  B,  or 
A  is  as  B.  Thus  the  wages  of  a  laboring  man  vary  as  the 
time  of  his  service.  We  say  that  the  interest  of  money  v/hich 
is  loaned  for  a  given  time,  is  proportioned  to  the  principal. 
But  a  proportion  contains  four  terms.  Here  are  only  two, 
the  interest  and  the  principal.  This,  then  is  an  abridged 
statement,  in  which  two  terms  are  mentioned  instead  of  foiu*. 
The  proportion  in  form  would  be  : 

As  any  given  principal,  is  to  any  other  principal ; 

So  is  the  mterest  of  the  former,  to  the  interest  of  the  latter. 

404.  In  many  mathematical  and  philosophical  investiga- 
tions, we  have  occasion  to  determine  the  general  relations 
of  certain  classes  of  quantities  to  each  other,  without  limiting 
the  inquiry  to  any  particular  values  of  those  quantities.  In 
such  cases,  it  is  frequently  sufficient  to  mention  only  two  of 
the  terms  of  a  proportion.  It  must  be  kept  in  mind,  how- 
ever, that  four  are  always  implied.  When  it  it  said,  for  in- 
stance, that  the  weight  of  water  is  proportioned  to  its  bulk, 
we  are  to  understand, 

That  one  gallon,  is  to  any  number  of  gallons  ; 
As  the  weight  of  one  gallon,  is  to  the  weight  of  the  given 
number  of  gallons. 

405.  The  character  en  is  used  to  express  the  proportion  of 
variable  quantities. 

Thus  A  en  B  signifies  that  A  varies  as  B,  that  is,  that  • 
A:a::B:b. 

The  expression  A  <ji  B  may  be  called  a  general  proportion. 


208  ALGEBRA. 

406.  Oae  quantity  is  said  to  vary  directly  as  another,  when 
the  one  increases  as  the  other  increases,  or  is  diminished  as 
the  other  is  dimmished,  so  that 

AcjiB,  that  is,  .^  :  a : :  B  :  b. 
The  interest  on  a  loan  is  increased  or  diminished,  in  pro- 
portion to  the  principal.     If  the  principal  is  doubled,  the  in- 
terest is  doubled ;  if  the  principal  is  trebled,  the  interest  is 
trebled,  &c. 

407.  One  quantity  is  said  to  vary  inversely  or  reciprocally 
as  another,  when  the  one  is  proportioned  to  the  reciprocal 
of  the  other ;  that  is,  w  hen  the  one  is  diminished,  as  the  other 
is  increased,  so  that 

Acji  —  that  is,  .^ :  a  : :  J-  :  1,  or  .^ :  a  :  :  6  :  K 
B  '  B     b' 

In  this  case,  if  A  is  ^eater  than  a,  B  is  less  than  b.  (Art. 
395.)  The  time  required  for  a  man  to  raise  a  given  sum,  by 
his  labor,  is  inversely  as  his  wages.  The  higher  his  wages, 
the  less  the  time. 

408.  One  quantity  is  said  to  vary  as  two  others  jointly,  when 
the  one  is  increased  or  diminished,  as  the  product  of  the  other 
two,  so  that 

A  (X  J5C,  that  is  A:  a:  :BC:  be. 

The  interest  of  money  varies  as  the  product  of  the  princi- 
pal and  time.  If  the  time  be  doubled,  and  the  principal 
doubled,  the  interest  will  be  four  times  as  great. 

409.  One  quantity  is  said  to  vary  directly  as  a  second,  and 
inversely  as  a  third,  when  the  fust  is  always  proportioned  to 
the  second  divided  by  the  third,  so  that 

A  <j)  — ,  that  is  .^  :  a  : :  —  :  -. 
C  C     c 

410.  To  understand  the  methods  by  which  the  statements 
of  the  relations  of  variable  quantities  are  changed  from  one 
form  to  another,  little  more  is  necessary,  than  to  rnake  an 
application  of  the  principles  of  common  proportion  ;  bearing 
constantly  in  mind,  that  a  general  proportion  is  only  an 
abridged  expression,  in  wiiich  two  terms'  are  mentioned  in- 
stead of  four.  When  the  deficient  terms  are  supplied,  the 
reason  of  the  several  operations  will,  in  most  cases,  be  appa- 
rent. 


VARIATION.  209 

411.  It  is  evident,  in  the  first  place,  that  the  order  of  the 
terms  in  a  general  proportion  may  be  inverted.    (Art  369.) 

If        A:  awB  :  6,  thatis,  if  Acj^B; 
Then  B  .hwJl:  a,  that  is,      B  ^  A. 

412.  If  one  or  both  of  the  terms  in  a  general  proportion, 
oe  multiplied  or  divided  by  a  constant  quantit}^,  the  proportion 
^vill  be  preserved. 

For  multiplying  or  dividing  one  or  both  of  the  terms  is  the 
same,  as  multiplying  or  dividing  analogous  terms  in  the  pro- 
portion expressed  at  length.   (Art.  382.  and  cor.  1.) 

If  A:  a'.:B  \h,         that  is,  if  A  a.  B, 

Then    mA  :  ma  :  :  B  :  b,      that  is,    mA  c/)  B, 
And      mA  :  ma  : :  mB :  mh,  that  is,    mA  en  niB,  &c. 

413.  If  both  the  terms  be  multiplied  or  divided  even  by 
a  variable  quantity,  the  proportion  will  be  preserved.  For 
this  is  equivalent  to  multiplying  the  two  antecedents  by  one 
quantity,  and  the  two  consequents  by  another.    (Art.  382.) 

If  A:  a::  B  :b,  that  is,  if  A  cj)  B ; 

Then  MA  :  ma  :  :  MB  :  mb,  that  is  MA  cr  MB,  &c. 

'Cor.  1.  If  one  quantity  varies  as  another,  the  quotient  of 
the  one  divided  by  the  other  is  constant.  In  other  words,  if 
the  numerator  of  a  fraction  varies  as  the  denominator,  the 
value  remains  the  same. 

If  A  :  a  :  :  B  :  b,  that  is,  if  A  cyi  /?, 

Then    -:?:::?:  ^  ::  1  :  1.     (Art.  128.) 
B     b      B    b  ^  ^ 

Here  the  third  and  fourth  terms  are  equal,  because  each  is 

equal  to  1.     Of  course  the  two  first  terms  are  equal  ;   (Art. 

395.)  so  that  if  A  be  increased  or  diminished  as  many  times 

as  B,  the  quotient  will  be  invariably  the  same. 

Cor.  2.  If   the  product  of  two  quantities  is  constant,  one 

varies  reciprocally  as  the  other. 

If  ^^:  a6  ::  1  :  l,then:i^  :  ^  :  :  i  :  1,  or  A  :  a:  :  1:1. 
B       b       B     b  B     b 

Cor.  '3.  Any  factor  in  one  term  of  a  general  proportion, 
may  be  transferred,  so  as  to  become  a  divisor  in  the  other ; 
and  V.  V. 

If  A  LnBC,  then  dividing  by  B,  —  cd  C.    (Art.  118.) 

B 

27 


210  ALGEBRA. 

If  ^  cr  J_,  then  mult,  by  C,  AC  <s>  L.     (Art.  159.) 
CD  ^     '  D      ^  ' 

414.  If  two  quantities  vary  respectively  as  a  third,  then 
one  of  the  two  varies  as  the  other.    ( A.rt.  384i) 

And    C:c::B:h\'^^^'''^'^  \  C  o^  B  ; 
Then  A:  a:  :  C  :  c,    that  is  Jl^  C. 

415.  If  two  quantities  vary  respectively  as  a  third,  their 
sum  or  difference  wiM  vary  in  the  same  manner.  (Art.  388.) 

If        A'.a-.'.B  :h}  ,.    ,  .      ,.  (  ^  c/)  5 

And    C:c::J5:6r^^'^^'^MCc^^; 
Then  A-{-C  :  a-\-c :  :  B  :  b,  that  is,  A-[-C  en  B, 
And    A-C:  a-c:  :B  :  b,  that  is,  A-ClpB. 

Cor.  The  addition  here  may  be  extended  to  any  number  of 
quantities  all  varying  alike.   (Art.  388.  cor.  1.) 

If  A  u^  B,  and  C  lp  B,  and  D  lti  J5,  and  E  en  B,  then 

(Jl^C-\-D+E)(jyB. 

415.  b.  If  the  square  of  the  sum  of  two  quantities,  vsyiea 
as  the  square  of  their  difference ;  then  the  sum  of  their  squares 
varies  as  their  product. 

If  {AJf-Bycf^{A  -  Bf]  then  JP-{-B'  c/)  AB. 
For  by  the  supposition, 

{A^Bf  :  (A -BY::  {a-^bf  :  {a -by. 
Expanding,  adding,  and  subtracting  terms.     (Arts.  217, 
and  389,  7.) 

2^+2^  :  4AB :  :  2a^4-26'  :  4ab. 
Or,  (Art.  382.) 

A'+B"  :  AB::  a''-\-b''  :  a^that  is,  A-\-B'  cr>AB. 

416.  The  terms  of  one  general  proportion  may  be  multi- 
plied or  divided  by  the  corresponding  terms  of  another. — 
(Art.  390.) 

If      A:a::B  :  b  )  ..^  •     ..  ^  A  <j^  B 
And  C:c::V  :  dT  '       (C^D; 


Then  AC  :  ac  :  :  BD  :  bd  that  is,  AC  ^  BD. 
Cor.  If  two  quantities  vary  respectively  as  a  third,  the  pro 
duct  of  the  two  will  vary  as  the  square  of  the  other. 


l^  /J/^?,] then  AC c/^B". 
And  C  cnB  ^ 


VARIATION.  211 

417.  If  any  quantity  vary  as  another,  o.ny  pmcer  or  root  of 
the  former  will  vary,  ag  a  like  power  or  root  of  the  latter. 
(Art.  391.) 

If        A  \  A.  '.B'.b,  that  is,  'liAc/iB, 

Then  A"",  a'^iiB"  ib"  that  is,      A"  cdB", 

1         i  i  1  XI 

And  A":a":  iB''  :  b",  that  is,  A''  <x  B". 

418.  In  compounding  general  proportions,  equal  factors  or 
divisors,  in  the  two  terms,  may  be  rejected.     (Art.  393.) 

If      A:a::B:b)  CAc/^B 

And  B  :b  :  :  C  :  c^    that  is,  if    {  B  en  C 
And  C:c:  :  D:dS  (  C  c/^  D 


Then  A  :  a:  :D  :  d,  that  is,  A O) D, 

Cor.  If  one  quantity  varies  as  a  second,  the  second,  as  a 
third,  the  third,  as  a  fourth,  &c.  then  the  first  varies  as  the 
last. 

If  A^BcfiCcjiD,  then  A  en  J). 

If  A  00  BcT)-,  then  Ac/>  — ;  that  is,  if  the  first  varies  di- 
C  C 

rectly  as  the  second,  and  the  second  varies  reciprocally  as  the 
third  ;  the  first  varies  reciprocally  as  the  third. 

419.  If  any  quantity  vary  as  the  product  of  two  others, 
and  if  one  of  the  latter  be  considered  constant,  the  first  will 
vary  as  the  other. 

If  JFc/)  LB,  and  if  B  be  constant,  then  W  er>  L. 

Here  it  must  be  observed  that  there  are  two  conditions ; 
First,  that  Ovaries  as  i\\e product  of  the  two  other  quantities; 
Secondly,  that  one  of  these  quantities  B  is  constant. 

Then,  by  the  condit  ions,  W :  w:  :  LB  :  IB ;  B  being  the 
same  in  both  terms. 

Divid.  by  the  constant  quantity  B,  W:  ic:  :L:l,  that  is  W(X  L. 
And  if  L  be  considered  constant,  WcdB. 

Thus  the  weight  of  a  board,  of  uniform  thickness  and  den- 
sity, varies  as  its  length  and  breadth.  If  the  length  is  given, 
the  weight  varies  as  the  breadth.  And  if  the  breadth  is  given, 
the  weight  varies  as  ,the  length. 


212  ALGEBRA. 

Cor.  The  same  principle  may  be  extended  to  any  number 
of  quantities.  The  weight  of  a  stick  of  timber,  of  given 
density,  depends  on  the  length,  breadth,  and  thickness.  If 
the  length  is  given,  the  weight  varies  as  the  breadth  and 
thickness.  If  the  length  and  breadth  are  given,  the  weight 
varies  as  the  thickness,  &c. 

If  W^LBT; 

Then  making  L  constant,  W  (ji  BT ; 

And  making  L  and  B  constant,  Wcr>  T; 

420.  On  the  other  hand,  if  one  quantity  depends  on  two 
others  ;  so  that  when  the  second  is  given,  the  first  varies  as 
the  third,  and  when  the  third  is  given,  the  first  varies  as  the 
second  ;  then  the  first  varies  as  the  product  of  the  other  two. 

If  the  weight  of  a  board  varies  as  the  length,  when  the 
breadth  is  given,  and  as  the.  breadth  when  the  length  is  giv- 
en :  then  if  the  length  and  breadth  both  vary,  the  weight  va- 
ries as  their  product. 

If       W(J)  L,  when  B  is  constant,  >  , ,        it/-      r>r 
A    1  Trr      r.      I        r  •  .     *   /  then  Wc/)BL. 

And  W  o)  i>,  when  L  is  constant,  5 

In  demonstrating  this,  we  have  to  consider,  two  variable  va- 
lues of  W;  one,  when  L  only  varies,  and  the  other,  when  L 
and  B  both  vary. 

Let  w'=  the  first  of  these  variable  values, 

And  to  =  the  other  ; 

So  that  W  will  be  changed  to  w',  by  the  varying  of       L  ; 

And  mo'  will  be  farther  changed  to  w,  by  the  varying  of  B. 
Then  by  the  supposition,  W:w'::L:l,  when  B  is  constant. 
And  w'  :w  :  :  B:  b,  when  B  varies. 


Mult,  correspond,  terms,  Ww' :  ww' : :  BL  :  bl.     (Art.  390.) 
Divid.  by  w'  (Art.  382.)    W:  id  :  :  BL  :bl,i.e.  W  (x>  BL. 

The  proof  may  be  extended  to  any  number  of  quantities. 

The  weight  of  a  piece  of  timber,  depends  on  its  length, 
breadth,  thickness  and  density.  If  any  three  of  these  are 
given,  the  weight  varies  as  the  other. 

This  case  must  not  be  confounded  with  that  in  Art.  416, 
cor.  In  that,  B  is  supposed  to  vary  as  A  and  as  C,  at  the 
same  time.  In  this,  B  varies  as  wJ,  only  when  C  is  constant, 
and  as  C,  only  when  Jl  is  constant.  It  cannot  therefore  vary 
ae  A  and  as  C  separately,  at  the  same  time. 


ARITHMETICAL  PROGRESSION.  213 

Art.  420.  b.  If  one  quantity  varies  as  another,  the  former  is 
equal  to  the  product  of  the  latter  into  seme  constant  quantity. 

If  ^:  B :  :  a:  b  ;  then,  whatever  be  the  value  of  a,  its  ratio 
to  b  must  be  constant,  viz.  that  of  A  :  B.  Let  this  ratio  l)e 
that  of  m:  I. 

Then  A:  B:  :  a:  b  :  :m:  I.  Therefore  A =mB ;  And  a=mb. 

Hence,  if  the  ratio  between  the  two  quantities  be  found 
for  any  given  value,  it  will  be  known  for  any  other  period  of 
their  increase  or  decrease.  If  the  interest  of  100  dollars  be 
to  the  principal  as  1  :  20  ;  the  interest  of  1000  or  10,000  will 
have  the  same  ratio  to  the  principal. 

421.  Many  writers,  in  expressing  a  general  proportion^  do 
not  use  the  term  vary,  or  the  character  which  has  here  been 
put  for  it.  Instead  of  A  cf>  B,  they  say  simply  that  A  is  as  B. 
See  Enfield's  Philosophy.  It  may  be  proper  to  observe,  al- 
so, that  the  word  given  is  frequently  used  to  distinguish  con- 
stant quantities,  from  those  which  are  variable  ;  as  well  as 
to  distinguish  known  quantities  from  those  which  are  un- 
known.    (Art.  17.) 


SECTION  XIV. 

ARITHMETICAL  AND  GEOMETRICAL  PROGRESSION. 

Art.  422.  QUANTITIES  which  decrease  by  a  common 
difference,  as  the  numbers  10,  8,  6,  4,  2,  are  in  continued 
arithmetical  proportion.  (Art.  372.)  Such  a  series  is  also 
called  a  progression,  which  is  only  another  name  for  continued 
proportion. 

It  is  evident  that  the  proportion  will  not  be  destroyed,  if 
the  order  of  the  quantities  be  inverted.  Thus  the  numbers 
2,  4,  6,  8,  10,  are  in  arithmetical  proportion. 


214  ALGEBRA. 

Quantities,  then,  are  in  arithmetical  progression,  when  they 
increase  or  decrease  by  a  common  difference. 

When  they  increase,  they  form  wliat  is  called  an  ascending 
series,  as  3,  5,  7,  9,  11,  &c. 

When  they  decrease,  they  form  a  descending  series,  as  1 1 , 
9,  7,  5,  &6. 

The  natural  numbers,  1,  2,  3,  4,  5,  6,  &c.  are  in  arithmet- 
ical progression  ascending. 

423.  From  the  definition  it  is  evident  that,  in  an  ascending 
series,  each  succeeding  term  is  found,  by  adding  the  common 
difference  to  the  preceding  term. 

If  the  first  term  is  3,  and  the  common  difference  2 ; 
The  series  is  3,  5,  7,  9,  11,  13,  &c. 
If  the  first  term  is  a,  and  the  common  difference  d ; 
Then  a-\-d  is  the  second  term,  a-{-2d-\-dz=a-]-od,  the  fourth, 
a-\-d-\-d=a-{-2d  the  3d,  a-\-Sd-]-d=a^4d  the  5th,  &c. 

12  3  4  5 

And  the  series  is  a,  a-\-d,  a-\-2d,  a-{-Sd,  a~\-4d,  Sic. 

If  the  first  term  and  the  common  diffference  are  the  same, 
the  series  becomes  more  simple.     Thus  if  a  is  the  first  term, 
and  the  common  difference,  and  n  the  number  of  terms. 
Then  a-\-a=2ais  the  second  term, 
2a-\-a=Sa  the  third,  &c. 
And  the  series  is  a,  2a,  3a,  4a, na. 

424.  In  a  descending  series,  each  succeeding  term  is  found, 
by  subtracting  the  common  difference  from  the  preceding  term. 

If  a  is  the  first  term,  and  d  the  common  difference,  the 

12  3  4  5 

series  is  a,  a  -  d,  a  -  2d,  a  -  Sd,  a  -  4d,  &c. 

Or  the  common  difference  in  this  case  may  be  considered 
VLS-d,  a  negative  quantity,  by  the  addition  of  which  to  any 
preceding  term,  we  obtain  the  following  term. 

In  this  manner,  we  may  obtain  any  term,  by  continued 
addition  or  subtraction.  But  in  a  long  series,  this  process 
would  become  tedious.  There  is  a  method  much  more  ex- 
peditious.    By  attending  to  the  series 

r         2  3  4  5 

a,  a-{-d,  a-\-2d,  a-{-Sd,  a-]-4d,  &c. 
it  will  be  seen,  that  the  number  of  times  d  is  added  to  a  is  one 
less  than  the  number  of  tlie  term. 


ARITHMETICAL  PROGRESSION.  215 

The  second  term  is  a-\-d,   i.  e.  a  added  let  once  d; 
The  third  is  a\-2d,  a  added  to  twice  d; 

The  fourth  is  a-\-3d,         a  added  to  thrice  d,  &c. 

So  if  the  series  be  continued, 

The  50th  term  will  be  a-^-idd 

The  100th  term  a^-dM 

If  the  series  be  descending,  the  100th  term  will  be  a — 9M. 

In  the  last  term,  the  number  of  times  d  is  added  to  a,  is 
one  less  than  the  number  of  all  the  terms.     If  then 

a=^the  first  term,  z=the  last,  n=:the  number  of  terms,  we 
shall  have,  in  all  cases,  z=a-\-{n-\)  xd;  that  is, 

425.  In  an  arithmetical  progression,  the  last  term  is  equal 
to  the  first, -\-  the  product  of  the  common  difference  into  the  number 
of  terms  less  one. 

Any  other  tenn  may  be  found  in  the  same  way.  For  the 
series  may  be  made  to  stop  at  any  term,  and  that  may  be 
considered,  for  the  time,  as  the  last. 

Thus  the  mth  term = a-|-  («i  - 1 )  X  ^. 

If  the  first  term  and  the  common  difference  are  the  samcy 

z=a-{-{n-l)a=a-\-na-ay  that  is,  z=na. 

In  an  ascending  series,  the  first  term  is,  evidently,  the  least, 
and  the  last,  the  greatest.  But  in  a  descending  series,  the 
first  term  is  the  greatest,  and  the  last,  the  least. 

426.  The  equation  z=a-\-{n -l)d  not  only  shews  the  value 
of  the  last  term,  but,  by  a  few  simple  reductions,  will  enable 
us  to  find  other  parts  of  the  series.  It  contains  four  different 
quantities, 

a,  the  first  term,  n,  the  number  of  terms,  and 

z,  the  last  term,  d,  the  common  difference. 

If  any  three  of  these  be  given,  the  other  may  be  found. 

1.  By  the  equation  already  found, 

z=a-\-{n -l)d=the  last  term, 

2.  Transposing  {n-l)d,  (Art.  173.) 

z-{n-l)d=a=the  first  term. 

3.  Transposing  a  in  the  1st,  and  dividing  by  n-1, 

z  —  a 
_.  =d=the  common  difference. 


216  ALGEBRA. 

4.  Transp.  a  in  the  1st,  dividing  by  d,  and  transp.  -1, 

z-a 
J    ■\-\=zn=the  number  of  terms. 

By  the  third  equation,  may  be  found  any  number  of  arith- 
meticnl  means,  between  two  given  numbers.  For  the  %chole 
number  of  terms  consists  of  the  two  extremes,  and  all  the 
intermediate  terms.  If  then  m=  the  number  of  means,  m-f- 
2=n,  the  whole  number  of  terms.  Substituting  m-j-2  for  n, 
in  the  third  equation,  we  have 

z-a 

~r=d.  the  common  difference. 


m-fl 

Prob.  1.  If  the  first  term  of  an  increasing  progression  is  7, 
the  common  difference  3,  and  the  number  of  terms  9,  what  is 
the  last  term?    Ans.  2r=a+(n-l)t/=7-f(9-l)x3  =  31. 

And  the  series  is  7,  10,  13,  16,  19,  22,  25,  28,  31. 

Prob.  2.  If  the  last  term  of  an  increasing  progression  is  60, 
the  number  of  terras  12,  and  the  common  difference  5,  what 
is  the  first  terni?    Ans.  a=z-{n-\)d=60 -{\2-l)x^=5' 

Prob.  3.  Find  6  arithmetical  means,  between  1  and  43. 
Ans.  The  common  difference  is  6. 

And  the  series,  1,  7,  13,  19,  25,  31,  37,  43. 

427.  There  is  one  other  inquiry  to  be  made  concerning  a 
series  in  arithmetical  progression.  It  is  often  necessary  to 
find  the  sum  of  all  the  terms.  This  is  called  the  summation  of 
the  series.  The  most  obvious  mode  of  obtaining  the  amount 
of  the  terms,  is  to  add  them  together.  But  the  nature  of 
progression  will  furnish  us  with  a  method  more  expeditious. 

It  is  manifest  that  the  sum  of  the  terms  will  be  the  same, 
in  whatever  order  they  are  written.  The  sum  of  the  ascend- 
ing series,  3,  5,  7,  9,  11,  is  the  same,  as  that  of  the  descend- 
ing series,  11,  9,  7,  5,  3.  The  sum  of  both  the  series  is, 
therefore,  ticice  as  great,  as  the  sum  of  the  terms  in  one  of 
them.  There  is  an  easy  method  of  finding  this  double  sum, 
and  of  course,  the  svim  itself  which  is  the  object  of  inquiry. 
Let  a  given  series  be  written,  both  in  the  direct,  and  in  the  in- 
verted order,  and  then  add  the  corresponding  terms  together. 


ARITHMETICAL  PROGRESSION.  217 

Take,  for  instance,  the  series  3,    5,    7,    9,  11, 

And  the  same  inverted  11,    9,    7,    5,    3. 


The  suras  of  the  terms  will  be         14,  14,  14,  14,  14. 
Take  also  the  series         a,  a-\-d,    a-\-2d,  a-f  3cZ,  a-|-4rf. 

And  the  same  inver.        «-|-4f/,  a-\-^d,  a-\-2d,  a-\-d,    a. 


The  sums  will  be         2a-i-4d,2a-\-'^d,2a-\-4d,2a-\-4d,2a-\-4d 
Here  we  discover  the  important  property,  that, 

428.  In  an  arithmetical  progression,  the  sum  of  the  ex- 
tremes IS  EQUAL  TO  THE  SUM  OF  ANY  OTHER  TWO  TERMS 
EQUALLY  DISTANT  FROM  THE  EXTREMES. 

In  the  series  of  numbers  above,  the  sum  of  the  first  and 
the  last  term,  of  the  first  but  one  and  the  last  but  one,  &c.  is 
14.  And  in  the  other  series,  the  sum  of  each  pair  of  corres- 
ponding terms  is  2a-\-4d. 

To  find  the  sum  of  all  the  terms  in  the  double  series,  we 
have  only  to  observe,  that  it  is  equal  to  the  sum  of  the  ex- 
tremes repeated  as  many  times  as  there  are  terms. 
The  sum  of  14,  14,  14,  14,  14=14x5. 

And  the  smn  of  the  terms  in  the  other  double  series  is 
-(2a-f4(/)x5. 

But  this  is  twice  the  sum  of  the  terms  in  the  single  series. 
If  then  \ve  put 

a=the  first  term,  n=:the  number  of  terms, 

2:= the  last,  s=the  sum  of  the  terms, 

we  shall  have  this  equation, 

s = —^r-  X  n.     That  is, 

429.  In  an  arithmetical  progression,  the  sum  of  all  the 

TERMS  IS  EQUAL  TO  HALF  THE  SUM   OF  THE    EXTREMES    MUL- 
TIPLIED INTO   THE  NUMBER  OF  TERMS. 

Prob.  What  is  the  simi  of  the  natural  series  of  numbers 
1,  2,  3,  4,  5,  &c.  up  to  10001 

a-fz              1  +  1000 
Ans.  5  =  -^  xn=  ^^ X  1000=500500. 

If  in  the  preceding  equation,  we  substitute  for  z,  its  value 
as  given  in  Art.  426,  we  have 


2a+(n-l)f^ 

28 


I.  5= 2 Xw. 


218  ALGEBRA. 

In  this,  there  are  four  different  quantities,  the  first  term  of 
the  series,  the  common  difference,  the  number  of  terms,  and 
the  sum  of  the  terms ;  any  three  of  which  being  given,  the 
fourth  may  be  found.  For,  by  reducmg  the  equation,  we 
have, 

2s-dn''-irdn     ,       ^ 

2.  a= ^ '  ^^^^  fi^^^  term. 

2s-2an 

3.  d= — o »  the  common  difjerence. 

n  —n  '^ 


4.  n='--^ ;  -{-  <^  g    -va  -f-  ^  ^i^^  number  of  terms. 

2d 

Ex.  1.  If  the  first  term  of  an  increasing  arithmetical  series 
is  3,  the  common  difference  2,  and  the  number  of  terms  20 ; 
what  is  the  sum  of  the  series  1  Ans.  440. 

2.  If  100  stones  be  placed  in  a  straight  line,  at  the  dis- 
tance of  a  yard  from  each  other;  how  far  must  a  person  tra- 
vel, to  bring  them  one  by  one  to  a  box  placed  at  the  distance 
of  a  yard  from  the  first  stone]     Ans.  5  miles  and  1300  yards. 

3.  What  is  the  sum  of  1 50  term^  of  the  series 

12        457 

3'  3'  ^'  3'  3'  ^'  3'  ^^- •  ^^^'  ^''"^^• 

4.  If  the  sum  of  an  arithmetical  series  is  1455,  the  least 
term  5,  and  the  number  of  terms  30  ;  what  is  the  common 
difference?  Ans.  3. 

5.  If  the  sum  of  an  arithmetical  series  is  567,  the  first 
term  7,  and  the  common  difference  2 ;  what  is  the  number 
of  terms'?  Ans.  21. 

6.  What  is  the  sum  of  32  terms  of  the  series 

1,  IJ,  2,  2J,  3,  &C.'?  Ans.  280. 

7.  A  gentleman  bought  47  books,  and  gave  10  cents  for 
the  first,  30  cents  for  the  second,  50  cents  for  the  third,  &:c. 
What  did  he  give  for  the  whole  1  Ans.  220  dollars,  90  cents 

8.  A  person  put  into  a  charity  box,  a  cent  the  first  day  of 
the  year,  two  cents  the  second  day,  three  cents  the  third  day, 
&c.  to  the  end  of  the  year.  What  was  the  whole  sum  for 
365  days  ?  *  Ans.  667  dollars,  95  cents. 


ARITHMETICAL  PROGRESSION.  219 

430.  Ill  the  series  of  odd  niunbers  1,  3,  5,  7,  9,  &:c.  con- 
tinued to  any  given  extent,  the  last  term  is  always  one  less 
than  twice  the  number  of  terms. 

For  z=a-{-(n  -  'i)d.  (Art.  425.)  But  in  the  proposed 
series  a=l,  and  d=2. 

The  equation,  then,  becomes  r==l  +  (n-l)  x2  =  2n-  1. 

431.  In  the  series  of  odd  numbers,  1,  3,  5,  7,  9,  &:c.  the 
sum  of  the  terms  is  always  equal  to  the  square  of  the  number  of 

"^tenns. 

For  s=\  (a-{-z)n.     (Art.  429.) 

But  here  a=l,  and  by  the  last  article,  2:=2?i-l. 
The  equation,  then,  becomes  s=^  {l-{-2n  -l)n=n^. 

Thus     1+3  =  4    ^ 

1 4-3-1-5  ==9    >  the  square  of  tlie  number  of  terms. 
l_j.34-5+7=16) 

432.  If  there  be  two  ranks  of  quantities  in  arithmetical 
progi'ession,  the  sums  or  differences  will  also  be  in  arithmetical 
progression. 

For  by  the  addition  or.  subtraction  of  the  corresponding 
terms,  the  ratios  are  added  or  subtracted.  (Art.  345.)  And 
by  the  nature  of  progression,  all  the  ratios  in  the  series  are 
equal.  Therefore  equal  ratios  being  added  to,  or  subtracted 
from,  equal  ratios,  the  new  ratios  thence  arising  will  also  be 
equal. 

To  and  from   3,    6,    9,  12,  15,  18,  21  ^  /-  3 

Add  and  sub.  2,    4,    6,    8,  10,  12,  14  f  .    .     )^ 

c     .  "Fin   1  ^   ort~o"-   on   q>  }  whose  ratio  is  <  T 

Smns  5,  10,  lo,  20,  2o,  30,  So  L  a  5 

Diff.  1,    2,    3,    4,    5,    6,    7  )  '  1 

433.  If  all  the  terms  of  an  arithmetical  progression  be  mul- 
tiplied or  divided  by  the  same  quantity,  the  products  or  quo- 
tients will  be  in  arithmetical  progression. 

For  by  the  multiplication  or  division  of  the  terms,  the  ratios 
are  multiplied  or  divided;  (Art.  344,)  that  is,  equal  quantities 
are  multiplied  or  divided  by  the  given  quantity.  They  will 
therefore  remain  equal. 

If  the  series  3,    5,    7,    9, 11,  &c.be  multiphed  by  4; 

The  prods,  will  be  12,  20,  28,  36,  44,  &c.  and  if  this  be  div.  by  2; 
Thequots.willbe   6, 10, 14,  18,  22,  &c. 


220  ALGEBRA. 

Problems  of  various  kind?,  in  aritlimetical  progression,  may 
be  solved,  by  stating  the  conditions  algebraically,  and  then 
reducing  the  equations. 

Prob.  1.  Find  four  numbers  in  arithmetical  progression, 
whose  sum  shall  be  56,  and  the  sum  of  their  squares  864. 

If  a:=the  second  of  the  four  numbers, 

And  y  =  the[r  common  difference: 
The  series  will  hex-y,  a:,  (c-{-y,  x-\-2y. 
By  the  conditions,      {x-y)-\-^-\-{^+y)-\-{x-\-2y)  =  56  > 
And  (a:-2/)'^+^'H-('^4-2/)'+(^+22/)'-=864  5 

That  is  4x+2y  =  56    > 

And  4x'^-\-4xy-\-6y'^z=S64 ) 

Reducing  these  equations,  we  have  a:  =12,  and  y  =  4. 

The  numbers  required,  therefore,  are        8,  1 2,  1 6,  and  20. 

Prob.  2.  The  sum  of  three  numbers  in  arithmetical  pro- 
gression is  9,  and  the  sum  of  their  cubes  is  153.  What  are 
the  numbers?  Ans.   1,  3,  and  5. 

Prob.  3.  The  sum  of  three  numbers  in  arithmetical  pro- 
gression is  1 5 ;  and  the  sum  of  the  squares  of  the  two  ex- 
tremes is  58.     What  are  the  numbers? 

Prob.  4.  There  are  four  numbers  in  arithmetical  progres- 
sion :  the  sum  of  the  squares  of  the  two  first  is  34 ;  and  the 
sura  of  the  squares  of  the  two  last  is  130.  What  are  the 
numbers?  Ans.  3,  5,  7,  and  9. 

Prob.  5.  A  certain  number  consists  of  three  digits,  which 
are  in  arithmetical  progression  ;  and  the  number  divided  by 
the  sum  of  its  digits  is  equal  to  26;  but  if  198  be  added  to 
it,  the  digits  will  be  inverted.     What  is  the  number? 

Let  the  digits  be  equal  to  x-y,  x,  and  x-\-y,  respectively. 
Then  the  number  =\00{x -y)^\0x^{x^ij)  =  lUx-99y. 

lUx-99y  ) 

By  the  conditions,  -c,— —  =26  f 

And  llla:-99i/4-198=100(a:+2/)+10a;+(a'-?/)   ) 

Therefore      x=S,y  =  l,  and  the  number  is  234. 

Prob,  6.  The  smu  of  the  squares  of  the  extremes  of  fom- 
numbers  in  arithmetical  progression  is  200 ;  and  the  sum  of 
the  squares  of  the  means  is  136.     What  are  the  numbers  ? 


GEOMETRICAL  PROGRESSION.  221 

Prob.  7.  There  are  four  numbers  in  arithmetical  progres- 
sion, whose  sum  is  28,  and  their  continual  product  585. 
What  are  the  numbers  ? 


GEOMETRICAL  PROGRESSION. 

434.  As  arithmetical  proportion  contirmed  is  arithmetical 
progression,  so  geometrical  proportion  continued  is  geometri- 
cal progression. 

The  numbers  64,  32,  16,  8,  4,  are  in  continued  geometri- 
cal proportion.     (Art.  372.) 

In  this  series,  if  each  preceding  term  be  divided  by  the 
common  ratio,  the  quotient  will  be  the  following  term. 
V  =  32,  and  \'  =  16,  and  \'  =  8,  and  |=4. 

If  the  order  of  the  series  be  inverted,  the  proportion  will 
still  be  preserved  ;  (Art.  399,)  and  the  connnon  divisor  will 
become  a  multiplier.     In  the  series 

4, 8,16,32,64,  &c.  4x2  =  8,  and  8x2  =  16,  and  16  x2=:32,iS:c. 

435.  Quantities  then  are  in  geometrical  progression, 

WHEN    THEY    INCREASE    BY    A    COMMON    MULTIPLIER,    OR   DE- 
CREASE BY  A  COMMON  DIVISOR. 

The  common  midtiplier  or  divisor  is  called  the  ratio.  For 
most  purposes,  however,  it  will  be  more  simple  to  consider 
the  ratio  as  always  a  midtiplier,  either  integral  or  fractional. 

In  the  series  64,  32,  16,  8,  4,  the  ratio  is  either  2  a  divisor, 
or  I  a  multiplier. 

To  investigate  the  properties  of  geometrical  progression, 
we  may  take  nearly  the  same  course,  as  in  arithmetical  pro- 
gression, observing  to  substitute  continual  multiplication  and 
division,  instead  of  addition  and  subtraction.  It  is  evident, 
in  the  first  place,  that, 

436.  In  an  ascending  geometrical  series,  each  succeeding 
term  is  found,  by  multiplying  the  ratio  into  the  preceding  term. 

If  the  first  term  is  a,  and  the  ratio  r, 

Then  axr=ar,  the  second  term,  ar^Xr=ctr^,  the  fourth, 
arxi'=ar'^,  the  third,  ar^Xr=ar^,  the  fifth,  &c. 

And  the  series  is  a,  ar,  ar'^,  ar^,  ar*,  ar%  &c. 

437.  If  the  first  term  arid  the  ratio  are  the  same,  the  pro- 
gression is  simply  a  series  of  powers. 


222  ALGEBRA. 

If  the  first  term  and  the  ratio  are  each  equal  to  r, 
Then  rxr—r\  the  second  term,     r^xr=r\  the  fourth, 
r^Xr=r^,  the  third,  r*Xr=r\  the  fifth. 

And  the  series  is  r,  r^,  r^,  r^,  r^,  r^,  &c. 

438.  In  a  descending  series,  each  succeeding  tenn  is  found 
by  dividing-  the  preceding  term  by  the  ratio,  or  multiplying 
by  the  fractional  ratio. 

If  the  first  term  is  ar^,  and  the  ratio  r, 

the  second  term  is  -— ,  or  ar^x^'j 

And  the  series  is  ar^,  ar%  ar\  ar%  ar\  ar,  a,  &c. 

If  the  first  term  is  a,  and  the  ratio  r, 

mi  .      .       a  (t  a 

Ihe  series  is  a,-,--,^-^,  &c.  or  a,  ar^\  ar~^,  &c. 

12         3  4  5  6 

By  attending  to  the  series  a,  ar,  m^,  ar%  ar\  ar^,  &c.  it  will 
be  seen  that,  in  each  term,  the  exponent  of  the  power  of  the 
ratio,  is  one  less,  than  the  number  of  the  term. 

If  then  a=the  first  term,  r=zthe  ratio, 

z  =  the  last,  n = the  number  of  terms ; 

we  have  the  equation  zz=ar''~\  that  is, 

439.  In  geometrical  progression,  the  last  term  is  equal  to  the 
product  of  the  first,  into  that  power  of  the  ratio  whose  index  is  one 
less  than  the  number  of  terms. 

When  the  least  term  and  the  ratio  are  the  same,  the  equa- 
tion becomes  zz=rr''-^—r''.     See  Art.  437. 

440.  Of  the  four  quantities  a,  z,  r,  and  n,  any  three  being 
given,  the  other  may  be  found.* 

1 .  By  the  last  article, 

2.  Dividing  by  f 


1 
} 

z 

— i=a=the  first  term. 


^-1 
3.  Dividing  the  1st  by  a,  and  extracting  the  root, 

=r=:the  ratio. 


w 


♦  See  Note  P. 


GEOMETRICAL   PROGRESSION.  223 

By  the  last  equation  may  be  foiird  any  number  of  geome- 
trical means,  between  two  given  numbers.  If  m=  the  num- 
ber of  means,  m-[-2  =  n,  the  whole  number  of  terms.  Substi- 
tuting m-[-2  for  n,  in  the  equation,  we  have 

J 

">-ri  =r,  the  ratio. 
a  I 

When  the  ratio  is  found,  tlie  means  are  obtained  by  con- 
tinued muUiphcation. 

Piob.   1.  Find  two  geometrical  means  between  4  and  256. 
Ans.  The  ratio  is  4,  and  the  series  is  4,  IG,  64,  256. 

Prob.  2.  Find  three  geometrical  means  between  -9  and  9. 

Ans,  i,  1,  and  3. 
441.  The  next  thing  to  be  attended  to,  is  the  rule  for  find- 
ing the  sum  of  all  the  terms. 

If  any  term,  in  a  geometrical  series,  be  multiplied  by  the 
ratio,  the  product  Avill  be  the  succeeding  term.  (Art.  436.) 
Of  course,  if  each  of  the  terms  be  multiplied  by  the  ratio,  a 
new  series  will  be  produced,  in  which  all  the  terms  except 
the  last  will  be  the  same,  as  all  except  the  first  in  the  other 
series.  To  make  this  plain,  let  the  new  series  be  written 
under  the  other,  in  such  a  manner,  that  each  term  shall  be 
removed  one  step  to  the  right  of  that  from  which  it  is  pro- 
duced in  the  line  above. 

Take,  for  instance,  the  series  2,  4,  8,  16,  32 

Multiplying  each  term  by  the  ratio,  Ave  have  4,  8,  16,  32,  64 

Here  it  will  be  seen  at  once,  that  the  four  last  terms  in  the 
upper  line  are  the  same,  as  the  four  first  in  the  lower  line. 
The  only  terms  which  are  not  in  both,  are  the  first  of  the  one 
series,  and  the  last  of  the  other.  So  that  when  we  subtract 
the  one  series,  from  the  other,  all  the  terms  except  these  two 
will  disappear,  by  balancing  each  other. 

If  the  given  series  is         a,  ar,  ar^,  ar\ ar"~\ 

Tlien  mult,  by  r,  we  have     ar,  ar',  ar\  ....  ar''~\  ar". 

Now  let  s=  the  sum  of  the  terms. 
Then                           s  =  a-f-  ar-[-  ar'^-\-  ar^,  ....-far""', 
And  mult,  by  r,  rs=     ar-\-ar'^-\-ar% -\-ar^~'^-{-af'. 

Subt'g  the  first  equation  from  the  second,  rs- 5= ar"- a 
'   And  dividing  by  (r-  1,)  (Art.  121.)  ^='^^^' 


224  ALGEBRA 

In  this  eqiiation,  ar^  is  the  last  term  in  the  new  series,  and 
is  therefore  the  product  of  the  ratio  into  the  last  term  in  the 
given  series. 

Therefore  5=*IlL^,  that  is. 

442.  The  sum  of  a  series  in  geometrical  progression  is 
found,  by  multiplying  the  last  term  into  the  ratio,  subtract- 
ing the  first  term,  ancL  dividing  the  remainder  by  the  ratio 
less  one. 

Prob.  1.  If  in  a  series  of  numbers  in  geometrical  pro- 
gression, the  first  temi  is  6,  the  last  term  1458,  and  the  ratio 
3,  what  is  the  sum  of  all  the  terms  ?     '  • 

Ans.  s='lZ^=^.}<l^^l^=USi. 
r-.\  3-1. 

Prob.  2.  If  the  first  term  of  a. decreasing  geometrical  se- 
ries is  I,  the  ratio  ^,  and  the  number  of  temis  5  ;  ^whd.t  is  the 
sum  of  the  series '?  -   -         v 

The  last  termznar^-'^iXGO^^T^o.     ^  i       *  ^        ^ 

And  the  sum  of  the  terms  =  ^  ^  t^— !  —  -— i.  ^ 

i-l  162  "" 

Prob.  3.  What  is  the  sum  of  tke  series,  1,  3,  9,  27,  &c.  to 
12  terms?  ^*  Aris.  265720. 

Prob.  4.  What  is  the  sum  of  ten  terms  of  the  series  1,  f, 

4^0,^                                              A         174075 
s,  „  ,*.&c.  •  Ans.  . 

*  59049 

443.  Quantities  in  geomeit^ical  progression  are  proportional 
to  their  differences. 

'""'  Let  the  series  be  a,  ar,  ar^,  ar^,  ar\  &c. 

By  the  nature  of  geometrical  progression, 

a  :  ar :  :  ar  \  ar^  :  \  ar"^  :  ar^ :  :  ar^  :  ar*,  &c. 

In  eafh  couplet  let  the  siiitecedent  be  subtracted  from  the 
consequeiit,  According  to  Ad:t.  389,  6. 

Then  a  :  ar:  :ar-a  :  ar"-^:-.  af  ~ar  :  ar^ - ar"^,  &c. 

•t 

That  is,  the  first  term  is  to  the  second,  as  the  difference 

between  the  first  and  second,  to  the  difiference  between  the 
second  and  third  ;  and  as  the  difference  between  the  second 
and  third,  to  the  difference  between  the  third  and  fourth,  &c. 


GEOMETRICAL  PROGRESSION.  gg^ 

Cor.  If  quantities  are  in  geometrical  progression,  their  di/^ 
ferences  are  also  in  geometrical  progression. 

Thus  the  numbers        3,    9,     27,     81,       243,  &c. 
And  their  differences        6,  18,     54,     162,  &:c.  are  in  geo- 
metrical progression^ 

444.  Several  quantities  are  said  to  be  in  Immnonical  progres- 
sion, when,  of  any  three  which  a^  contiguous  in  the  series, 
the  first  is  to  the  la«t,  as  the  difference  between  the  two  first, 

to  the  difference  between  the  two  last.     See  Art.  400. 

» 

Thps  the  numbers  60,  30,  20,  15,  12,  10,  are  in  harmoni- 

cai  progression. 

Fof€0  :2a:: 60- 30:  30^  20,  And  20:  12::  20- 15:  15-12 
AndSe:15::30-20:2d-15,Andl5^  10::15-12:  12-10. 

Prec^ms  in  geometriccj  progression,  fnay  be  solved,  as  in 
othei^pafts  of  algebra, 'by  the  reductii^h  of  equations. 

Prob.  I  Find  three  numDCfs  in  geoijietrical  progression, 
such  that  theh"  sum  shall'  be  14,  ans^  the  sum  of  their 
squares  84.   , 

^,  '  L^t  the  three  noiiAbers  be  x,  y,  and  z. 

By  the  ciindftions,  x  :  y  :  :  y  :  z,    or  xz=y^  ^ 

And  •  x-\-y^z -14} 

And  *  ■      r'-^y'j^z'=8i) 

Reducing  these  equations,  we  find  the  numbers  required 
to  be  2,  4  and  8. 

Prob.  2.  There  are  three  numbers  in  geometrical  progres- 
sion whose  product  i^  64,  and  the  sum  of  their  cubes  is  584. 

What  are  the  numbers  1  * 

""  «• 

If  X  be  the  fiist  term,  and  ^  tl)p  common  ratio ;    the  series 

will  be  X,  xy,  xy^.       \^ 

By  the  conditions,     *      xx^X^^,         or  x^y^=64,    > 
And  *  :^+a:^t/='+xy^=584.  5 

These  equations  reduced  givl  a:=2,  and  y=2. 
The  numbers  required,  therefore  are,  2,  4  and  8. 

Prob.  3.^There  are  three  numbers  in  geometrical  progres- 
sion :  The  sum  of  the  first  and  last  is  52,  and  the  square  of 
the  mean  is  100.    What  are  the  numbers  1  Ans.  2,  10, and  50. 

29 


226  ALGEBRA. 

Prob.  4.  Of  four  numbers  in  geometrical  progression,  the 
sum  of  the  two  firstis  15,  and  the  sum  of  the  two  last  is  60. 
What  are  the  numbers  1 

Let  the  series  be  x,  ly,  xy\  xy'^  ;  and  the  numbers  will  be 
found  to  l3e  5,  10,  20,  and  40. 

Prob.  5.  A  gentleman  divided  210  dollars  among  three 
servants,  in  such  a  manner,  that  their  portions  vv'eve  in  geo- 
metrical progression  ;  and  the  first  had  90  dollars  more  than 
the  last.      How  much  had  each  1 

Prob.  6.  There  arc  three  numbers  in  geometrical  progres- 
sion, the  greatest  of  which  exceeds  the  least  by  15  ;  and  the 
difference  of  the  squares  of  the  greatest  and  the  least,  is  to 
the  sum  of  the  squares  of  all  the  three  numbers  as  5  to  7. 
What  are  the  numbers'?  Ans.  5,  10,  and  20. 

Prob.  7.  There  are  four  numbers  in  geometrical  progres- 
sion, the  second  of  which  is  less  than  tlie  fourth  by  24  ;  and 
the  sum  of  the  extremes  is  to  the  sum  of  the  means,  as  7  to  3. 
What  are  the  numbers  1  Ans.  1,  3,  9,  27. 


SECTION  XV. 

INFINITES  AND  INFINITESIMALS.* 


Art.  445.  THE  word  infinite  is  used  in  different  senses. 
The  ambiguity  of  the  term  has  been  the  occasion  of  much 
perplexity.  It  has  even  led  to  the  absurd  supposition  that 
propositions  directly  contradictory  to  each  other,  may  be 
mathematically  demonstrated.  These  apparent  contradic- 
tions are  owing  to  the  fact,  that  what  is  proved  of  infinity 


♦  Locke's  Essays,  Book  2,  Chap.  17.  Berkeley's  Analyst.  Preface  to  Mac- 
laurin's  Fluxions.  Newton's  Princip.  Saunderson's  Algebra,  Art.  388. 
Mansfield's  Essays.     Emerson's  Algebra,  Prob.  73.     Buffier. 


MATHEMATICAL  INFINITY.  227 

when  understood  in  one  particular  manner,  is  often  thought 
to  be  true  also,  when  the  term  has  a  ver}'  ditierent  significa- 
tion. Tlie  two  meanings  are  hisensibly  shifted,  the  one  for 
the  other,  so  that  tlie  proposition  which  is  really  demonstra- 
ted, is  exchanged  for  another  Vv-hich  is  ialse  and  alisurd.  To 
prevent  mistakes  of  this  nature,  it  is  important  that  the  dif- 
ferent meanings  be  carefully  distinguished  from  each  other. 

446.  Infinite,  in  the  highest,  and  perhaps  the  most  proper 
sense  of  the  word,  is  that  lohlch  is  so  great,  that  nothing  can  be 
added  to  it,  or  supposed  to  be  added. 

In  this  sense,  it  is  frequently  used  in  speaking  of  moral  and 
metaphysical  subjects.  Thus,  by  infinite  wisdom  is  meant 
that  which  will  not  admit  of  the  least  addition.  Infinite  power 
is  that  which  cannot  possibly  be  increased,  even  in  supposi- 
tion. This  meaning  of  infinity  is  not  applicable  to  the  ma- 
thematics. That  which  is  the  subject  of  the  mathematics  is 
quantity;  (Art.  1.)  such  quantity  as  may  be  conceived  of  by  the 
human  mind.  But  no  idea  can  be  formed  of  a  quantity  so 
great  that  nothing  can  be  supposed  to  be  added  to  it.  In  this 
sense,  an  infinite  number  is  mconccivable.  We  may  increase 
a  number  by  continual  addition,  till  v>'e  obtain  one  that  shall 
exceed  any  limits  which  v\'e  please  to  assign.  By  this,  how- 
ever, we  do  not  arrive  at  a  number  to  v/Jiich  nothing  can  be 
added  ;  but  only  at  one  that  is  beyond  any  limits  which  we 
have  hitherto  set.  Farther  additions  may  be  made  to  it  with 
the  same  ease,  as  those  by  which  it  has  already  beeiit  in- 
creased so  far.  It  is  therefore  not  infinite,  in  the  sense  in 
which  the  term  has  now  been  explained.  It  is  absurd  to 
speak  of  the  greatest  possible  number.  No  number  can  be 
imagined  so  great  as  not  to  admit  of  being  made  greater. 
We  must  therefore  look  for  another  meaning  of  infinity,  be- 
fore we  can  apply  it,  v.'ith  proprietj-,  to  the  mathematics. 

447.  A  MATHEMATICAL  QUANTITY  IS  SAID  TO  BE  INFINITE, 
WHEN  IT  IS  SUPPOSED  TO  EE  INCREASED  BEYOND  ANY  DETER- 
MINATE LIMITS. 

By  deteiTiiinate  limits  are  meant  such  as  can  be  distinctly 
stated.*  In  this  sense,  the  natural  series  of  numbers,  1,  2,  3,  4, 
5,  &c.  may  be  said  to  be  infinite.  For,  if  any  number  be  men- 
tioned ever  so  great,  another  may  be  supposed  still  greater. 

The  two  significations  of  the  word  infinite  are  liable  to  be 
confounded,  because  they  are  in  several  points  of  view  the 

*  See  Not«  Q, 


228  ALGEBRA. 

same.  The  higher  meaning  inchides  the  lower.  That  which 
is  so  great  as  to  admit  of  no  addition,  must  be  beyond  any 
determinate  Hmits.  But  the  lower  does  not  necessarily  imply 
the  higher.  Though  number  is  capable  of  being  increased 
beyond  any  specified  limits  ;  it  will  not  follow,  that  a  number 
can  be  found  to  which  no  farther  additions  can  be  made. 
The  two  infinites  agree  in  this,  that  according  to  each,  the 
things  spoken  of  are  great  beyond  calculation.  But  they 
differ  widely  in  another  respect.  To  the  one,  nothing  can  be 
added.     To  the  other,  additions  can  be  made  at  pleasure. 

448.  In  the  mathematical  sense  of  the  term,  there  is  no 
absurdity  in  supposing  one  infinite  greater  than  another. 

We  may  conceive  the  numbers  2  2  2  2  2  2  2,  &c. 

4  4  4  4  4  4  4,  &c. 

to  be  each  extended  so  far  as  to  reach  round  the  globe,  or  to 
the  most  distant  visible  star,  or  beyond  any  greater  boundary 
which  can  be  mentioned.  But  if  the  two  series  be  equally 
extended,  the  amount  of  the  one  will  be  ticice  as  great  as  the 
other,  though  both  be  infinite. 

So  if  the  series         a-{-  a^-j-  "^+  ^^H"  "^  ^^• 
and  9rt4-9a*-[-9rt^-f  9a*4-9«',  &c. 

be  extended  together  beyond  any  specified  limits,  one  will  be 
nine  times  as  great  as  the  other.  But  it  w^ould  be  absurd  to 
suppose  one  quantity  greater  than  another,  if  the  latter  were 
akeady  so  great  that  nothing  could  be  added  to  it. 

449.  An  infinite  number  of  terms  must  not  be  mistaken  for 
an  infinite  quantity.  The  terms  may  be  extended  beyond 
any  given  limits,  when  the  amount  of  the  whole  is  a  finite 
quantity,  and  even  a  small  one.  If  we  take  half  of  a  unit ; 
tnen  half  of  the  remainder  ;  half  of  the  remaining  half,  &c. 
we  Bhall  have  the  series 

in  which  each  succeeding  term  is  half  of  the  preceding  one. 
Let  the  progression  be  continued  ever  so  far,  the  sum  of  all 
the  terms  can  never  exceed  a  unit.  For,  by  the  supposition, 
there  is  still  a  remainder  equal  to  the  last  term.  And  this 
remainder  must  be  added,  before  the  amount  of  the  whole 
can  be  equal  to  a  unit. 

So  a-ht-f  «+tV+^+it'  ^c.  can  never  exceed  8. 


MATHEMATICAL  INFINITY.  229 

450.  When  a  quantity  is  diminished  till  it  becomes 
less  than  any  determinate  quantity,  it  is  called  an 
INFINITESIMAL. 

Thus,  in  a  series  of  fracti  ms  lo'  rio' loVo' tttsso'  &^c.  a 
unit  is  first  divided  into  ten  parts,  then  into  a  hundred,  a 
thousand,  &c.  One  of  these  parts  in  each  succeeding  term 
is  ten  times  less  than  in  the  preceding.  If  then  the  progres- 
sion be  continued,  a  portion  of  a  unit  may  be  obtained  less 
than  any  specified  quantity.  This  is  an  infinitesimal,  and  in 
mathematical  language,  is  said  to  be  infinitely  small.  By  this, 
however,  we  are  not  to  understand  that  it  cannot  be  made 
less.  The  same  process  that  has  reduced  it  below  any  hmit 
which  we  have  yet  specified,  may  be  continued,  so  as  to  di- 
minish it  still  more.  And  however  far  the  progression  may 
be  carried,  we  shall  never  arrive  at  a  point  where  we  must 
necessarily  stop. 

451.  In  the  sense  now  explained,  mathematical  quantity 
may  be  said  to  be  infinitely  divibible ;  that  is,  it  may  be  sup- 
posed to  be  so  divided,  that  the  parts  shall  be  less  than  any 
determinate  quantity,  and  the  number  of  parts  greater  than 
any  given  number. 

In  the  series  f^o,  twj  -nnnrj  i  oooo>  &c.  a  unit  is  di\aded 
into  a  greater  and  greater  number  of  parts,  till  they  become 
infinitesimals,  and  the  number  of  them  infinite,  that  is,  such 
a  number  as  exceeds  any  given  number.  But  this  does  not 
prove  that  we  can  ever  arrive  at  a  division  in  which  the  parts 
shall  be  the  least  possible  or  the  number  of  parts  the  greatest 
possible. 

452.  One  infinitesunal  may  be  less  than  another. 
The  series,  j^j  tIit,  -nrW  rrrloTr,  &c.  > 

AnH  3         3  3  3         Rrr'     ( 

-^^^^  To'  ToiT>  TffiTtr'  To?!7nr»  y^^.  ) 

may  be  carried  on  together,  till  the  last  term  in  each  becomes 
infinitely  small ;  and  yet  one  of  these  terms  Avill  be  only  half 
as  great  as  the  other.  For  the  denominators  being  the  same, 
the  fractions  will  be  as  their  numerators,  (Art.  360,  cor.  2,) 
that  is,  as  6  :  3,  or  2  :  1. 

Two  quantities  may  also  be  divided,  each  into  an  infinite 
number  of  parts,  using  the  tenn  infinite  in  the  mathematical 
sense,  and  yet  the  parts  of  one  be  more  numerous  than  those 
of  the  other. 

1  he  series   i  oj  to o;  riyxnT}  too o o>  ^^• 

1  I 1 

nnr;  iTnnr)  nnnnr> 


....  . .  .jnnry  &c.  ) 


230  ALGEBRA. 

may  both  be  infinitely  extended ;  and  yet  a  imit  in  the  last 
series,  is  divided  into  four  times  as  many  parts  as  in  the  first. 
But  if,  by  an  infinite  number  of  parts  were  meant  such  a 
number  as  could  not  be  incrersed,  rt  would  be  absurd  to  sup- 
pose the  divisions  of  any  quantity  to  be  still  m>ore  numerous.* 

453.  For  all  practical  purposes,  an  infinitesimal  may  be 
considered  as  absolutely  nothing.  As  it  is  less  than  any  de- 
terminate quantity,  it  is  lost  even  in  numerical  calculations. 
In  algebraic  processes,  a  term  is  often  rejected  as  of  no  value, 
because  it  is  infinitely  small. 

It  is  frequently  expedient  to  admit  into  a  calculation,  a 
small  error,  or  what  is  suspected  to  be  an  error.  It  may  be 
difficult  either  to  avoid  tlie  objectionable  part,  or  to  ascertain 
its  exact  value,  or  even  to  determine,  without  a  long  and 
tedious  process,  whether  it  is  reallv  an  error  or  not.  But  if  it 
can  be  shown  to  be  infiiiitely  small,  it  is  of  no  account  in 
practice,  and  may  be  retained  or  rejected  at  pleasure. 

It  is  impossible  to  find  a  decimal  which  sball  be  exactly 
equal  to  the  vulgar  fraction  ^.  Dividing  the  numerator  by 
the  denominator,  we  obtain  in  the  first  place  i^o-  This  is 
nearly  equal  to  f.     But  i^^o  is  nearer,  ^^q,  still  nearer,  &c. 

The  error,  in  the  first  instance,  is  3^. 

For  i%-{-s\=^o+3\=^^=h 

In  the  same  manner  it  may  be  shown,  that 

the  difference  between   \  ]  "  \  'too  *  ^^'       0 

I  i  and  .333,  is  g^j  &c. 

If  the  decimal  be  supposed  to  be  extended  beyond  any  as- 
signable limit,  the  difference  still  remaining  will  be  infinitely 
small.  As  this  error  is  less  than  any  given  quantity,  it  is  of 
no  account,  and  may  be  considered  in  calculation  as  nothing. 

454.  From  the  preceding  example  it  will  be  seen,  that  a 
quantity  may  be  continually  coming  nearer  to  another,  and 
yet  never  reach  it.  The  decimal  0.3333333,  &c.  by  repeated 
additions  on  the  right,  may  be  made  to  approximate  continu- 
ally to  I,  but  can  never  exactly  equal  it.  A  difference  will 
always  remain,  though  it  may  become  infinitely  small. 


♦  Soe  Note  R. 


MATHEMATICAL  INFINITY.  ggl 

When  one  quantity  is  tlius  made  to  approach  continually 
to  another,  without  ever  passin^^  it ;  the  latter  is  called  a 
limit  of  the  former.  The  fraction  |  isa  limit  of  the  decimal 
0.666  &c.  indefinitely  continued. 

455.  Though  an  infinitesimal  is  of  no  account  of  itself^ 
yet  its  effect  on  other  quantities  is  not  always  to  he  disre- 
garded. 

When  it  is  a  factor  or  divisor,  it  may  have  an  important 
influence.  It  is  necessary,  therefore,  to  attend  to  the  rela- 
tions which  infinites,  infinitesimals,  and  finite  quantities  have 
to  each  other.  As  an  infinitesimal  is  less  than  any  assigna- 
ble quantity,  as  it  is  next  to  nothing,  and,  in  practice,  may  be 
considered  as  nothing,  it  is  fieqiienLiy  represented  by  0. 

An  infinite  quantity  is  expressed  by  the  character  00 

456.  As  an  infinite  quantity  is  incomparably  greater  than 
a  finite  one,  the  alteration  of  the  former,  by  an  addition  or 
subtraction  of  the  latter,  may  be  disregarded  in  calculation. 
A  single  grain  of  sand  is  greater  in  comparison  with  the 
whole  earth,  than  any  finite  quantity  in  comparison  with  one 
v/hich  is  infinite.  If  therefore  infinite  and  finite  quanti- 
ties are  connected  by  the  sign  -[-  or  -,  the  latter  may  be  re- 
jected as  of  no  comparative  value.  For  the  same  reason,  if 
finite  quantities  and  infinitesimals  are  connected  by  -J-  or  -  , 
the  latter  may  be  expunged. 

457.  But  if  an  infinite  quantity  be  multiplied  by  one  which 
is  finite,  it  will  be  as  many  times  increased  as  any  other  quan- 
tity would,  by  the  same  multiplier. 

If  the  infinite  series  2  2  2  2  2  2  &c.  be  multiphed  by  4  ; 

The  product  v/ill  be  8  8  8  8  8  8  &c.  four  times  as  great  as 
the  multiphcand.     See  Art.  448. 

458.  And  if  an  infinite  quantity  be  divided  by  a  finite  quan- 
tity, it  will  be  altered  in  the  same  manner  as  any  other  quan- 
tity. 

If  the  infinite  series  6  6  6  6  6  6  6  6    &c.  be  divided  by  2 ; 
The  quotient  will  be  3  3  3  3  3  3  3  3  &c.  half  as  great  as 
the  dividend. 

459.  If  a  jitiite  quantity  be  multiplied  by  an  infinitesimal, 
the  product  will  be  an  infinitesimal  ;  that  is,  putting  z  for  a 
finite  quantity,  and  0  for  an  infinitesimal,  (Art.  455. 

zxO=0. 


232  ALGEBRA. 

If  the  multiplier  were  a  vnit,  the  product  would  be  equal 
to  the  multiplicand.  (Art.  90.)  If  the  multiplier  is  less  than 
a  unit,  the  product  is  proportionally  less.  If  then  the  multi- 
plier is  infinitely  less  than  a  unit,  the  product  must  be  nifi- 
nitely  less  than  the  multiplicand,  that  is,  it  must  be  an  infi- 
nitesimal. Or,  if  an  infinitesimal  be  considered  as  abso- 
lutely nothing,  then  the  product  of  z  into  nothing  is  nothing. 
(Art.  112.) 

460.  On  the  other  hand,  if  a  finite  quantity  be  divided  by 
an  infinitesimal,  the  quotient  will  be  infinite. 

-=  oo. 
0 

For,  the  less  the  divisor,  the  greater  the  quotient.  If  then 
the  divisor  be  infinitely  small,  the  quotient  will  be  infinitely 
great.  In  other  words,  an  infinitesimal  is  contained  an  infi- 
nite number  of  times  in  a  finite  quantity.  This  may,  at  first, 
appear  paradoxical.  But  it  is  evident,  that  the  quotient  must 
increase  as  the  divisor  is  diminished. 

Thus  6—3  =  2,  6-f-0.03  =  200, 

64.0.3  =  20,  6-^0.003  =  2000,  &c. 

If  then  the  divisor  be  reduced,  so  as  to  become  less  than 
any  assignable  quantity,  the  quotient  must  be  greater  than 
any  assignable  quantity. 

461.  If  a  finite  quantity  be  divided  by  an  infinite  quantity, 
the  quotient  will  be  an  infinitesimal. 

1=0. 

For  the  greater-  the  divisor,  the  less  the  quotient.  If  then, 
while  the  dividend  is  finite,  the  divisor  be  infinitely  great,  the 
quotient  will  be  infinitely  small. 

It  must  not  be  forgotten,  that  the  expressions  infinitely  great 
and  infinitely  small,  are,  all  along,  to  be  understood  in  the 
mathematical  sense  according  to  the  definitions  in  Arts.  447, 
and  450. 


DIVISION-  23S 


SECTION  XVI. 

DIVISION  BY  COMPOUND  DIVISORS,  GREATEST 
COMMON  MEASURE. 

Art.  462.  IN  the  section  on  division,  the  case  in  which 
the  divisor  is  a  compound  quantity  was  omitted,  because  the 
operation  in  most  instances,  requires  some  knowledge  of  the 
nature  of  powers ;  a  subject  which  had  not  been  previously 
explained. 

Division  by  a  compound  divisor  is  performed  by  the  fol- 
lowing rule,  which  is  substantially  the  same,  as  the  rule  for 
division  in  aritlimetic  ; 

To  obtain  the  first  term  of  the  quotient,  divide  the  first 
term  of  the  dividend,  by  the  first  term  of  the  divisor  ;* 

Multiply  the  whole  divisor,  by  the  term  placed  in  the  quo- 
tient ;  subtract  the  product  from  a  part  of  the  dividend  ;  and 
to  the  remainder  bring  down  as  many  of  the  following  terms, 
as  shall  be  necessary  to  continue  the  operation  : 

Divide  again  by  the  first  term  of  the  divisor,  and  proceed 
as  before,  till  all  the  terms  of  the  dividend  are  brought  down 

Ex.  1.  Divide  ac-\-hc-\-ad-{-hd,  by  a-\-h. 

a-\-h)ac-\-bc-\-ad-^hd{c-\-d 

ac-{-hc^  the  first  subtrahend. 


ad-\-hd 

ad-\-bd,  the  second  subtrahend. 


Here  ac,  the  first  term  of  the  dividend,  is  divided  by  a, 
the  first  term  of  the  divisor,  (Art.  116.)  which  gives  c  for  the 
first  term  of  the  quotient.  Multiplying  the  whole  divisor  by 
this,  we  have  ac-\-bc  to  be  subtracted  from  the  two  first 
terms  of  the  dividend.  The  two  remaining  terms  are  then 
brought  down,  and  the  first  of  them  is  divided  by  the  first 

*  See  Note  S. 
30 


234  ALGEBRA. 

term  of  the  divisor  as  before.  This  gives  d  for  the  second 
term  of  the  quotient.  Then  multiplying  the  divisor  by  dy 
we  have  ad-\-bd  to  be  subtracted,  which  exhausts  the  whole 
dividend  without  leaving  any  remainder. 

The  rule  is  founded  on  this  principle,  that  the  product  of 
the  divisor  into  the  several  parts  of  the  quotient,  is  equal  to 
the  dividend.  (Art.  115.)  Now  by  the  operation,  the  pro- 
duct of  the  divisor  into  the  first  term  of  the  quotient  is  sub- 
tracted from  the  dividend ;  then  the  product  of  the  divisor 
into  the  second  term  of  the  quotient ;  and  so  on,  till  the  pro- 
duct of  the  divisor  into  each  term  of  the  quotient,  that  is, 
the  product  of  the  divisor  into  the  ichole  quotient,  (Art.  100.) 
is  taken  from  the  dividend.  If  there  is  no  remainder,  it  is 
evident  that  this  product  is  equal  to  the  dividend.  If  there 
15  a  remainder,  the  product  of  the  divisor  and  quotient  is  equal 
to  the  whole  of  the  dividend  except  the  remainder.  And  this 
remainder  is  not  included  in  the  parts  subtracted  from  the 
dividend,  by  operating  according  to  the  rule. 

463.  Before  beginning  to  divide,  it  will  generally  be  ex- 
pedient to  make  some  preparation  in  the  arrangement  of  the 
terms. 

The  letter  which  is  in  the  first  term  of  the  divisor,  should 
be  in  the  first  term  of  the  dividend  also.  And  the  powers  of 
this  letter  should  be  arranged  in  order,  both  in  the  divisor 
and  in  the  dividend ;  the  highest  power  standing  first,  the 
next  highest  next,  and  so  on. 

Ex.  2.  Divide  2a'b-{-¥-\-2ab^-\-a%  by  a'^+6^+a6. 

Here,  if  we  take  a^  for  the  first  term  of  the  divisor,  the 
other  terms  should  be  arranged  according  to  the  powers  of  a, 
thus, 

a'24.a6+6«)a34-2a^6+2rtft'-f6'(a+6 
a'J^a'^bJrab' 


a''b-]-ab''-{.b' 
a^bJrab'+b' 


In  these  operations,  particular  care  will  be  necessary  in  the 
management  of  negative  quantities.  Constant  attention  must 
be  paid  to  the  rules  for  the  signs  in  subtraction,  multiplica- 
tion and  division.     (Arts.  82,  105,  123.) 


DIVISION.  235 

Ex.  3.  Divide  2ax -  2a^x - Sarxy-\-Ga^x-\-axy  ■-xyhy2a-y. 

If  the  terms  be  arranged  according  to  the  powers  of  a, 
they  will  stand  thus  ; 

2a  -  y)6a^x  -  Sa^xy  -  2a'^x-\-axy-\-2ax  -  xy{Mx  -  ax-\-x. 
Qa^x-Sa^xy 


2a'^x-\-ax^ 
2d^x-{-axy 


*-  -^2ax-xy 
-\-2ax  -  xy 


464.  In  multiplication,  some  of  the  terms,  by  balancing 
each  other,  may  be  lost  in  the  product.  (Art.  110.)  These 
may  re-appear  in  division,  so  as  to  present  terms,  in  the 
course  of  the  process,  different  from  any  which  are  in  the 
dividend. 

Ex.  4. 

a-\.x)(^-{-a^{a^  -  ax-{-x'^ 
(j^-[.a^x 


*  _  g^x-\-^ 
-  a^x  -  ax^ 


*       ax^-\-x^ 


Ex.  5. 


o*-  2aa;+2ar^)a^+4x^(a2+2aa:+2x* 
a*_2a'a;+2aV 


*4-2ft^a;-2aV+4a;* 
-|-2a'a;-4aV-|-4ax3 


*     4-2aV-4aa;3+4a;* 

If  the  learner  will  take  the  trouble  to  multiply  the  quo- 
tient into  the  divisor,  in  the  two  last  examples,  he  will  find, 


336  ALGEBRA. 

in  the  partial  products,  the  several  terms  which  appear  in  the 
process  of  dividing.  But  most  of  them,  by  balancing  each 
other,  are  lost  in  the  general  product. 

Ex.  6.  Divide  a'-{-(i''-{-a''b-^ab-Jf-Sac-\-Sc,  by  a+l. 

Quotient.     a^-^ab-\-Sc. 
Ex.  7.  Divide  a-\-b  -c-ax-  bx-\-cx,  by  a-\-b  -  c. 

Quotient.     \  -  x. 

Ex.  8.  Divide  2a'-  lSa'x+Ua''x''-8ax^-{-2x\  by  20,^  -  ax 
4-ar^  Quotient,     a'' -  6ax-\-2x:' 

465.  When  there  is  a  remainder  after  all  the  terms  of  the 
dividend  have  been  brought  down,  this  may  be  placed  over 
the  divisor  and  added  to  the  quotient,  as  in  arithmetic, 

Ex.  9. 

a+b)ac+bc+ad-{-bd+x(c-{-d-^  J—. 

a-{-b 
ac-^-bc 


*    *  ad^bd 
ad-\-bd 


* 


Ex.  10. 
d  -  h)ad  -  ah+bd  -  bh+y{a+b-\ — ^ 
ad  -  ah 


*     bd-bh 
bd-bh 


It  is  evident  that  a-\-b  is  the  quotient  belonging  to  the 
whole  of  the  dividenu,  excepting  the  remainder  y.  (Art.  562.) 

And_^  is  the  quotient  belonging  to  this  remainder.  (Art. 
o  — a 


1«4.) 


DIVISION.  237 

Ex.  11.  Divide  eax-{-2xy  -Sab  -by-\-3ac-{-cy-{-h,  by  Sa+y. 

Quotient.  2.r  -  6-(-c-f- __^ 
Sa-\-y' 

Ex.  12.  Divide. a^6-3a-2+2«6-6rt- 46+22,  by  6-3. 

Quotient.  a'4-2a  -  44- JL2, 
^     ■      ^6-3 

Ex.  13.  See  Art.  283. 

a-\-\/b)ac-\-c\/b-\-aA^d-{-y\/bcl(c-\-\/d. 
ac-\-c\/b 


a\/d-\-\/bd 
a\/d^A^bd 


Ex.  14.  Divide  a-{-\/y-{-ar\/y-{-ry,  by  a+V.V- 

Quotient.      \-^ry\/y. 

1 5.  Divide  x^  -  3rta;^-|-3a-a:  -  a^,  by  a:  -  a. 

16.  Divide  2i/^-  19?/-^+26t/  -  17,  by  ?/  -8. 

17.  Dividea:«-l,bya:-l. 

18.  Divide  4x'  -  0x--\-6x  -  3,  by  2x''-{-^x  -  1 . 

19.  Divide  a'-{-4a''b-]-Sb\  by  rt+26. 

20.  Divide  a;'  -  «V4-2rt'a;  -  a\  by  a;'  -  ax-{-a'. 

466.  A  regular  series  of  quotients  is  obtained,  by  dividing 
the  difference  of  the  powers  of  two  quantities,  by  the  differ- 
ence of  the  quantities.     Thus, 

(2/'-«')-^(2/-«)=2/4-«j 

(f-'a')^(y-ct)=y'+ayJra% 

iy'  -  «')-T-(2/  -  «)  =y'-\-ay'Jrei'y+a\ 

iy'  -  a')-r-(2/  -  «)  =2/'+«y'H-«Y+«'y+«S 

&c. 

Here  it  will  be  seen,  that  the  index  of  i/,  in  the  first  term 
of  the  quotient,  is  less  by  1,  than  in  the  dividend  ;  and  that 
it  decreases  by  1,  from  the  first  tenn  to  the  last  but  one  : 

While  the  index  of  a,  increases  by  1,  from  the  second  term 
to  the  last,  where  it  is  less  by  1,  than  in  the  dividend. 


238  ALGEBRA. 

This  may  be  expressed  in  a  general  formula,  thus, 

{y--a-)^{y-a)=y—'-\-ay—K...+a—'y+a--\ 

To  demonstrate  this,  we  have  only  to  multiply  the  quo- 
tient into  the  divisor.   (Art.  115.) 

All  the  terms  except  two,  in  the  partial  products,  will  be 
balanced  6y  each  other  ;  and  will  leave  the  general  product 
the  same  as  the  dividend. 

Mult.  ?/^-|-a?/^+ay+a'?/+a* 
Into    y  -  a 

y'+ay'+aY+a,V+a'y 
-  ay"^  -  ahf-  a^y^  -  a^y  -  a^ 

Product.!/'      *       *       *       *     -a' 


Mult.  1/— i-l-at/— 2+ay-3 _|_a---2/+a'"-  ' 

Into   y -a 


-ay""-^-  ary""-^ -  a"*"  Y  -  oT'^y  -  a" 


Prod.  I/"*     * 


466.  b.  In  tlie  same  manner  it  may  be  proved,  that  the  dif- 
ference of  the  powers  of  two  quantities,  if  the  index  is  an 
even  number,  is  divisible  by  the  sum  of  the  quantities.  That 
is,  as  the  double  of  every  nmnber  is  even  ; 

(i/^-  a"")-i-(i/+a)  =2/'^'"-^  -  ay"^-^ +a'""-')j  -  a'^'\ 

And  the  sum  of  the  powers  of  two  quantities,  if  the  index 
is  an  odd  number,  is  divisible  by  the  sum  of  the  quantities. 
That  is,  as  2m-\-\  is  an  odd  number  ; 

For  in  each  of  these  cases,  the  product  of  the  quotient  and 
divisor,  is  equal  to  the  dividend. 
Thus, 

(!/'-a')-T-(!/+«)=2/-«> 

(y'  -  «^)H-(2/+«)  =y'  -  af+a)j  -  a\ 

iy'  -  a*) -T-(!/-f  «)  =  y'  -  «y+«y  -  ay+a*y  -  a',  &c. 


COMMON  MEASURE.  239 

And, 

(2/'+«') -J-(2/+«)  =y'-  ^y'-^(^V  -  ay^aY  -  a'y-\-a\  &c. 

^  GREATEST  COMMON  MEASURE. 

466.  c.  The  Greatest  Common  Measm-e  of  two  quantities, 
may  be  found  by  the  following  rule  ; 

Divide  one  of  the  quantities  by  the  other,  and  the 
preceding  divisor  by  the  last  remainder.  till  nothing 

remains;  THE  LAST  DIVISOR  WILL  BE  THE  GREATEST  COMMON 
MEASURE. 

The  algebraic  letters  are  here  supposed  to  stand  for  whole 
numbers.  In  the  demonstration  of  the  rule,  the  following 
principles  must  be  admitted. 

1.  Any  quantity  measures  itself,  the  quotient  being  1. 

2.  If  two  quantities  are  respectiv^ely  measured  by  a  thii'd, 
their  sum  or  difference  is  measured  by  that  third  quantity. — 
If  b  and  c  are  each  measured  by  d,  it  is  evident  that  b-\-c, 
and  b-c  are  measured  by  d.  Connecting  them  by  the  sign-j- 
.:)r  -,  does  not  affect  their  capacity  of  being  measured  by  d. 

Hence,  if  b  is  measured  by  d,  then  by  the  preceding  pro- 
position, b-{-d  is  measured  by  d. 

3.  If  one  quantity  is  measured  by  another,  any  midtiple 
of  the  former  is  measured  by  the  latter.  If  b  is  measured 
by  d,  it  is  evident  that  b-\-b,  36,  46,  nb,  &c.  are  measured 
hyd. 

Now  let  D=ihe  greater,  and  t^=the  less  of  two  algebraic 
quantities,  whether  simple  or  compound.  And  let  the  pro- 
cess of  dividing,  according  to  the  rule  be  as  follows  : 

d)D{q 
dq 

r)d(q' 

T')r(q"' 
r'q" 


240  ALGEBRA. 

In  which  9,  q^  q'^,  are  the  quotients,  from  the  successive 
divisions  ;  and  r,  r\  and  0  the  remainders.  And  as  the  divi- 
dend is  eqnal  to  the  product  of  the  divisor  and  quotient  added 
to  the  remainder, 

D  =  dq-\-r,  and  dz=rq'-\-r^. 

Then,  as  the  last  divisor  r'  measures  r  the  remainder  being  0, 
it  measures  (2,  and  3,)  rq'-\-r'=d, 
and  measures  dq-{-r=D, 

That  is,  the  k\st  divisor  r'  is  a  common  measure  of  the  two 
given  quantities  D  and  d. 

It  is  also  their  greatest  common  measure.  For  every  com- 
mon measure  of  D  and  d,  is  also  (3,  and  2)  a  measure  of 
D—  dq  =  r  ;  and  every  common  measure  of  d  and  r,  is  also  a 
measure  of  d-rq^=r\  But  the  greatest  measure  of  r^  is 
itself.  This,  then,  is  the  greatest  common  measure  of  D 
and  d. 

The  demonstration  will  be  substantially  the  same,  what- 
ever be  the  number  of  successive  divisions,  if  the  operation 
be  continued  till  the  remainder  is  nothing. 

To  find  the  greatest  common  measure  of  three  quantities  ; 
first  find  the  greatest  common  measure  of  two  of  them,  and 
then,  the  greatest  common  measme  of  this  and  the  ^third 
quantit)^  If  the  greatest  common  measure  of  D  and  d  be 
r\  the  greatest  common  measure  of  r^  and  c,  is  the  greatest 
common  measure  of  the  three  quantities  D,  d,  and  c.  For 
every  measure  of  r^,  is  a  measure  of  D  and  d ;  therefore  the 
greatest  common  measure  of  r'  and  c,  is  also  the  greatest 
common  measure  of  D,  d,  and  c. 

The  rule  may  be  extended  to  any  number  of  quantities. 

466.  d.  There  is  not  much  occasion  for  the  preceding 
operations,  in  finding  the  greatest  common  measure  of  sim- 
ple algebraic  quantities.  For  this  piupose,  a  glance  of  the 
eye  will  generally  be  sufficient.  In  the  application  of  the 
rule  to  compound  quantities,  it  will  frequently  be  expedient 
to  reduce  the  divisor,  or  enlarge  the  dividend,  in  conformit) 
with  the  following  principle  ; 

The  greatest  common  measure  of  two  quantities  is  not  altered, 
by  multiplying  or*dividing  either  of  them  by  any  quantity  which 
is  not  a  divisor  of  the  other,  and  which  contains  no  factor  which 
is  a  divisor  of  the  other. 

The  common  measure  of  ab  and  ac  is  a.  If  either  be 
multiplied  by  d,  the  common  measure  of  abd,  and  ac,  or  of 


COMMON  MEASURE.  241 

ab  and  acd,  is  still  a.  On  the  other  hand,  if  ab  and  acd  are 
the  given  quantities,  the  common  measure  is  a;  and  if  acd 
be  divided  by  d,  the  common  measure  of  ab  and  ac  is  a. 

Hence  in  finding  the  common  measure  by  division,  the 
divisor  may  often  be  rendered  more  simple,  by  dividing  it  by 
some  quantity,  which  does  not  contain  a  divisor  of  the  divi- 
dend. Or  the  dividend  may  be  multiplied  by  a  factor,  which 
does  not  contain  a  measure  of  the  divisor. 

Ex.  1.  Find  the  greatest  common  measure  of 
6a^+llaa:+3ar^,  and  Qa^J^'^ax -Sx\ 

6a''-\-7ax  -  3i:')6a'+l  1  ax-ir^x-{l 


Dividing  by  2x)4ax-{-6x'^ 

2a-{-Sx)6a'-\-7ax  -  ^x\Sa  -  x 


2ax  -  So:' 
lax-^x" 


After  the  first  division  here,  the  remainder  is  divided  by 
2a:,  which  reduces  it  to  %a~\-^x.  The  division  of  the  pre- 
ceding divisor  by  this,  leaves  no  remainder.  Therefore  2a-f- 
3a:  is  the  common  measure  required. 

2.  What  is  the  greatest  common  measure  of  a^  -  b^x,  and 
x--{-2bx-{-b''1  Ans.    x-\-b. 

3.  What  is  the  greatest  common  measure  of  cx-^-x"^,  and 
a'^c-\-d^x  1  Ans.  c-\-x. 

4.  What  is  tlie  greatest  common  measure  of  Sx^  -  24a:  -  9, 
and2a:3-  iq^.q^  Ans.  3:^-83:- 3. 

5.  What  is  the  greatest  common  measure  of  a*  -  b\  and 
a'-6V]  Ans,  a''-b'\ 

6.  What  is  the  greatest  common  measure  of  ar^-  1,  and 
xy-\-y1  Ans.  x-\-\. 

7.  What  IS  the  greatest  common  measure  of  x^  -  a^,  and 


31 


242  ALGEBRA. 

8.  What  is  the  greatest  common  measure  of  a^-ab-  2b\ 
and  a"- Sab +2b''l 

9.  What  is  the  greatest  common  measure  of  a*  -  x\  and 

a^-a^x-ax'-^-x'l 

10.  What  is  the  greatest  common  measure  of  a?  -  ab\  and 


SECTION  XVII. 


INVOLUTION  AND  EXPANSION  OF  BINOMIALS.* 

Art.  467.  THE  manner  in  which  a  binomial,  as  well  as 
any  other  compound  quantity,  may  be  involved  by  repeated 
multiplications,  has  been  shown  in  the  section  on  powers. 
(Art.  213.)  But  when  a  high  power  is  required,  the  opera- 
tion becomes  long  and  tedious. 

Tliis  has  led  mathematicians  to  seek  for  some  general  prin- 
ciple, by  which  the  involution  may  be  more  easily  and  expe- 
ditiously performed.  We  are  chiefly  indebted  to  Sir  Isaac 
Newton  for  the  method  which  is  now  in  common  use.  It  is 
founded  on  what  is  called  the  Binomial  Theorem,  the  inven- 
tion of  which  Avas  deemed  of  such  importance  to  mathemati- 
cal investigation,  that  it  is  engraved  on  his  monument  in 
Westminster  Abbey 

468.  If  the  binomial  root  be  a+6,  we  may  obtain,  by  mul- 
tiphcation,  the  following  powers.      (Art.  213.) 

*  Simpson's  Algebra,  Sec.  15.  Simpson's  Fluxions,  Art.  99.  Euler's  Alge- 
bra, Sec.  2.  Chap.  10.  Manning's  Algebra.  Saunderson's  Algebra,  Art. 
380.  Vince's  Fluxion's,  Art.  33.  Warmg's  Med.  Anal.  p.  415.  Lacroix's 
Algebra,  Art.  13.^.  Do.  Comp.  Art.  70.  Lond.  Phil.  Trans.  1795,  1816,  and 
1817.     Woodhouse's  Analytical  Calculation. 


INVOLUTION  OF  BINOMIALS.  243 

(a-^by=a'-{-2ab-i-h- 

(a-l^by=a'+4a'b-\-6a'b''+4ab'-^b' 
la+by-a'-]-oa'b-{-l0a'b^-\-l0arb'-{-5ab'Jrb%  &c. 

By  attending  to  this  series  of  powers,  we  shall  find,  that 
the  exponents  preserve  an  invariable  order  through  the  whole. 
This  will  be  very  obvious,  if  we  take  the  exponents  by  them- 
selves, unconnected  with  the  letters  to  which  they  belong. 

T    *u  *i  ,  ( of  a  are  2,  1,  0 

In  the  square,  the  exponents  <    f  />  a  •    0   12 

In  the  cube,  the  exponents  |  "J^  °  ^J^  J  f '  ^'  ^ 

T    *u     >i*u  4i  ^       ^  of  a  are  4,  3,  2,  1,  0 

In  the  4th  power,  the  exponents     |  ^^  ^  ^^.^  ^'^  ^^'  ^^  3'  4 


& 


c. 


Here  it  will  be  seen  at  once,  that  the  exponents  of  a  in  the 
first  term,  and  of  b  in  the  last,  are  each  equal  to  the  index  of 
the  power  ;  and  that  the  sum  of  the  exponents  of  the  two  let- 
ters is  in  every  term  the  same.     Thus  in  the  fourth  power, 

C  in  the  first  term,  is  4-|-0=4 
The  sum  of  the  exponents   <  in  the  second,        3-j-l  =4 

(  in  the  third,  2-i-2  =  4,&c. 

It  is  farther  to  be  observed,  that  the  exponents  of  a  regu- 
larly decrease  to  0,  and  that  the  exponents  of  b  increase  from 
0.  That  this  will  universally  be  the  case,  to  whatever  ex- 
tent the  involution  may  be  carried,  will  be  evident,  if  we  con- 
sider, that  in  raising  from  any  power  to  the  next,  each  term 
is  multiplied  both  by  a  and  by  b. 

Thus  {a-\-by=a''-{-2ab-{-b^ 
Mult,  by  a+6 

[of  a  in  each  term. 

a^-{-2a'^b-\-ab'^.       Here  1  is  added  to  the  exp. 
a'b-^2ab''-{-b\s     Here  1  is  added  to  the 

[exp.  of  Z)  in  each  term. 

(a+by=a^-\-Sa:'b-{-Sab''-{-bK 

If  the  exponents,  before  the  multiplication,  increase  and 
decrease  by  1,  and  if  the  multiplication  adds  1  to  each,  it  is 
evident  they  must  still  increase  and  decrease  in  the  same 
manner  as  before. 


244  ALGEBRA. 

469.  If  tlien  a-\-h  be  raised  to  a  power  whose  exponent  is  n. 
The  exp's  of  a  will  be  n,  «  -  1 ,  ?i  -  2,  .  .  .  .  2,  1,     0 ; 
And  the  exp's  of  h  will  be  0,  1,        2,  .  .  .  .  ?i  -  2,  n  -  1,  n. 

The  terms  in  which  a  power  is  expressed,  consist  of  the 
letters  with  their  exponents,  and  tlie  co-efficients.  Setting  aside 
the  co-efficients  for  the  present,  we  can  determine,  from  the 
preceding  observations,  the  letters  and  exponents  of  any 
power  whatever. 

Thus  the  eighth  power  of  a-j-^,  when  written  without  the 
co-efficients,  is 

a«  -f  a'b  +  a'b''  +  a'b'  +  a'b'  +  (v¥  +  a'b'-}-  aV  +  b\ 

And  tlie  nth  power  of  a^b  is, 

a'«_[_a'-'64-rt"-262 a'6"-^-f  rt6"-»4-6". 

470.  The  number  of  terms  is  greater  by  1,  than  the  index 
of  the  power.  For  if  the  index  of  the  power  is  n,  a  has,  in 
different  terms,  every  index  from  n  down  to  1  ;  and  there  is 
one  additional  term  which  contains  only  b.     Thus, 

The  square  has  3  terms,         The  4th  power,  5, 
The  cube  4,  The  5th  power,  G,  &c. 

471.  The  next  step  is  to  find  the  co-efficients.  This  part 
ot  the  subject  is  more  complicated. 

In  the  series  of  powers  at  the  beginning  of  Art.  468,  the 
co-efficients,  taken  separate  from  the  letters  are  as  follows  ; 
In  the  square,  1,  2,  1,  whose  sum  is  4=2'' 

In  the  cube,  1,3,  3/ 1,  8  =  2' 

In  the  4th  power,     1,  4,  6,  4,  1,  16  =  2* 

In  the  5th  power,  1,  5, 10, 10,  5,  1,  32  =  2^ 

The  order  which  these  co-efficients  observe  is  not  obvious, 
like  that  of  the  exponents,  upon  a  bare  inspection.  But  they 
will  be  found  on  examination  to  be  all  subject  to  the  follow- 
ing law ; 

472.  The  co-efficient  of  the  first  term  is  1  ;  that  of  the 
second  is  equal  to  the  index  of  the  power ;  and  universally, 
if  the  co-effixient  of  any  term,  be  multiplied  by  the  index  of 
the  leading  quantity  in  that  term,  and  divided  by  the  index  of 
the  following  quantity  increased  by  1,  it  will  give  the  co- 
efficient of  the  succeeding  term.* 

*  See  Note  T. 


INVOLUTION  OF  BINOMIALS.  245 

Of  the  two  letters  in  a  term,  the  first  is  called  the  leading 
quantity,  and  the  other  the  following  quantity.     In  the  ex 
amples  which  have   been   given   in   this  section,  a  is    the 
leading  quantity,  and  h  the  following  quantity. 

It  may  frequently  be  convenient  to  represent  the  co-effi- 
cients in  the  several  terms,  by  the  capital  letters,  *^,  B,  C,  &c. 

The  ntli  power  of  a-\-h,  without  the  co-efficients,  is 
ar-\-ar-'h-\-a''-^h''-^ar-W-\-ar-'b\  &c.   (Art.  469.) 
And  the  co-efficients  are, 
Jl=n,  the  co-efficient  of  the  second  term  ; 

n—\ 

B  =znX 5  of  the  third  term  ; 

2 

C  =  n X ^—  X^^~^>      of  tlie  fourth  term  ; 

/)^7J,X— -x'— X^^,  of  the  ^^///i  term;   &c.. 
A&  o  4 

The  regular  manner  in  which  these  co-efficients  are  de 
rived  one  from  another,  will  be  readily  perceived. 

473.  By  recurring  to  the  numbers  in  Art.  471,  it  will  be 
seen,  that  the  co-efficients  first  increase,  and  then  decrease,  at 
the  same  rate  ;  so  that  they  are  equal,  in  the  first  term  and 
the  last,  in  the  second  and  last  but  one,  in  the  third  and  last 
but  two ;  and  universally,  in  any  two  terms  equally  distant 
from  the  extremes.  The  reason  of  this  is,  that  (^a-{-by  is  the 
same  as  (b-j-a)" ;  and  if  the  order  of  the  terms  in  the  bino- 
mial root  be  changed,  the  w^hole  series  of  terms  in  the  oower 
will  be  inverted. 

It  is  sufficient,  then,  to  find  the  co-efficients  of  half  the 
terms.     These  repeated  will  serve  for  the  whole. 

474.  In  any  power  of  {a-\-b,)  the  sum  of  the  co-efficients 
is  equal  to  the  number  2  raised  to  that  power.  See  the  list 
of  co-efficients  in  Art.  471,  The  reason  of  this  is,  that,  ac- 
cording to  the  rules  of  multiplication,  when  any  quantity  is 
involved,  the  letters  are  multiplied  into  each  other,  and  the 
co-efficients  into  each  other.  Now  the  co-efficients  of  a-\-b 
being  l-[-l  =  2,  if  these  be  involved,  a  series  of  the  power? 
of  2  will  be  produced. 

475.  The  principles  which  have  now  l>een  explained  may 
mostly  be  comprised  in  the  following  general  theorem,  called 


246  ALGEBRA. 

THE  BINOMIAL  THEOREM. 

The  index  of  the  leading  quantity  of  the  power 
of  a  binomial,  begins  in  the  first  term  with  the  in- 
dex of  the  power,  and  decreases  regularly  by  1. 
The  index  of  the  following  quantity  begins  with  1 
in  the  second  term  and  increases  regularly  by  1. 
(Art.  468.) 

The  CO-EFFICIENT  of  the  first  term  is  1  ;  that 
of  the  second  is  equal  to  the  index  of  the  power  ; 
and  universally,  if  the  co-efficient  of  any  term  be 
multiplied  by  the  index  of  the  leading  quantity  in 
that  term,  and  divided  by  the  index  of  the  follow- 
ing quantity  increased  by  1,  it  will  give  the  co-ef- 
ficient of  the  succeeding  term.    (art.  472.) 

In  algebraic  characters,  the  tlieorem  is 

(a-f6)''=a"-f wXfl"-'  b-\-nX^^a''-~b%  &c. 

It  is  here  supposed,  that  the  terms  of  the  binomial  have  no 
Other  co-efficients  or  exponents  than  1.  Other  binomials  may 
be  reduced  to  this  form  by  substitution. 

Ex.  1 .  What  is  the  6th  power  of  x-{-y  ? 

The  terms  without  the  co-efficients,  are 
A  x%  x'y\  xy,  xY,  xy\  y\ 
And  the  co-efficients,  are 

1      r.     6x5     15x4     20x3    ^    . 
'      '    -2"'     ~3-'    ~4-'     '     • 
that  is  1,    6,    15,         20,  15,      6,  1. 

Prefixing  these  to  the  several  terms,  we  have  the  power 
required ; 

a;6_|_6a,''i/4- 1 5a;y-f  20ary+ 1 5xY-\-Qxy'-\.y\ 

2.  {d+hy=  d'Jf-^d'h-\-l0(Ph^-{-\0d'h'-^5dh'+h'. 

3.  What  is  the  nth  power  of  b-\-y  1 

Ans.  h''-JrM"-'y+Bb''-Y+Cb''-Y-{-I)b"-y,  &c. 

That  is,  supplying  the  co-efficients  which  are  here  repre- 
sented by  A,  B,  C,  &c.   (Art.  472.) 

6n+nx6"-V4-»iX^ixt''-y,  &c. 


rNVOLLJTION  OF  BINOJVftALS.  247 

4.  What  is  the  5th  power  of  x'-{-Sy^  1 
Substituting  a  for  x^,  and  6  for  Sy"^,  we  have 

{a-i-by=a'-lr5a'b-irl0aW~^\0a'b'+5ab'+b\ 

And  restoring  the  vakies  of  a  and  b, 

5.  What  is  the  sixth  power  of  {Sx-\-2y)  1 

Ans. 

729a;' +291 6x^2/+  4S60xY-]-4S20xY-\-^l60x'y'-\-^76xy' 
+Q4y\ 

476.  A  residual  quantity  may  be  involved  in  the  same 
manner,  without  any  variation,  except  in  the  signs.  By  re- 
peated muUiplications,  as  in  Art.  213,  we  obtain  the  follow- 
ing powers  of  (a-b.) 

(a-by=a''-2ab-\-b\ 

{a-by=a'-Sa%Jr^ab''-  b\ 

(a  -  by=a'  -  4a'b+6a^b'  -  4ab'-{-b\  &c. 

By  comparing  these  with  the  like  powers  of  {a-{-b)  in  Art. 
468,  it  will  be  seen,  that  there  is  no  difference  except  in  the 
signs.  There,  all  the  terms  are  positive.  Here,  the  terms 
which  contain  the  odd  powers  of  b  are  negative.  See  Art. 
218. 

The  sixth  power  of  (x  -  y)  is 

x^  -  6x'y-{-l5xY  -  20xy-\-15xY  -  6xy^-\-y^. 

The  nth  power  of  {a  -  b)  is 

a-" - M^-'b+Ba'^-^b'-  Ca'^-'b',  &c. 

477.  When  one  of  the  terms  of  a  binomial  is  a  unit,  it  is 
generally  omitted  in  the  power,  except  in  the  first  or  last 
term  ;  because  every  power  of  1  iS  1,  (Art.  209.)  and  this 
when  it  is  a  factor,  has  no  effect  upon  the  quantity  with 
which  it  is  connected.     (Art.  90.) 

Thus  the  cube  of  {x-{-l)  is         a;'4-3a;^Xl+3a;Xl'+l', 
Which  is  the  same  as  x^-{-Sx^-\-Sx-\-l. 

The  insertion  of  the  powers  of  1  is  of  no  vise,  unless  it 
be  to  preserve  the  exponents  of  both  the  leading  and  the  fol- 
lowing quantity  in  each  term,  for  the  purpose  of  finding  the 
co-efiicients.  But  this  will  be  unnecessary,  if  we  bear  in 
mind,  that  th«  sum  of  the   two  exponents,  in  each  term,  is 


248  ALGEBRA. 

equal  to  the  index  of  tlie  power.  (Art.  468.)  So  that,  if  we 
have  the  exponent  of  the  leading  quantity,  we  may  know 
that  of  the  following  quantity,  and  v.  v. 
Ex.  1.  The  sixth  power  of  (1  -?/)  is 

1  _  6i/-f  1 5y^  -  20y^Jr  1  ^v'  -  ^v'+V'- 

478.  From  the  comparatively  simple  manner  in  which  the 
power  is  expressed,  when  the  first  term  of  the  root  is  a  unit, 
is  sug-gested  the  expediency  of  reducing  other  binomials  to 
this  form. 

The  quotient  of  {a-\-x)  divided  by  a  is  [  1-}--] .    This  mul- 
tiplied into  the  divisor,  is  equal  to  the  dividend ;  that  is, 
{a+x)^ax  (l+-] therefore  {a+xy=a''x  (!+-)"• 

By  expanding  the  factor  i  1+-  i  ,  we  have 


-al 

(a+xy=:a'^X  f  1+-)"=«''X  U+A^+B'^+C^)  &c. 
\       aj  \  a        or       a^  J 

479.  When  the  index  of  the  power  to  which  any  binomial 
is  to  be  raised  is  a  positive  whole  number,  the  series  will  temii- 
nate.  The  number  of  terms  will  be  limited,  as  in  all  the 
preceding  examples. 

For,  as  the  index  of  the  leading  quantity  continually  de- 
creases by  one,  it  must,  in  the  end,  becoine  0,  and  then  the 
series  will  break  off. 

Thus  the  5th  term  of  the  fourth  power  of  a-\-x  is  x%  or 
aV,  a*^  being  commonly -omitted,  because  it  is  equal  to  1. 
(Art.  207.)  If  we  attempt  to  continue  the  series  farther,  the 
co-efficient  of  the  next  term,  according  to  the  rule,  will  be 

^^^-0.      (Art.  112.)   And  as  the  co-efficients  of  all  suc- 
ceeding terms  must  depend  on  this,  they  will  also  be  0. 

480.  If  the  index  of  the  proposed  power  is  negative,  this 
can  never  become  0,  b}^  the  successive  subtractions  of  a  unit. 
The  series  will,  therefore,  never  terminate;  but  like  many  de- 
cimal fractions,  may  be  continued  to  any  extent  that  is  de- 
sired. 


INVOLUTION  OF  BINOMIALS.  249 

Ex.  Expand  into  a  series =(a+t/)~*. 

The  terms  without  the  co-efficients,  are 

a-\  a-%  a-y,  a-y,  a-'y'^  ^^' 

The  co-efficient  of  the  2d  term  is  -  2,  of  the  4th"^^~-=-4. 

o 

Of  the  tliird,-Zl>lZ?zz.+3,of  the  5th  zl><Z:5  =+5. 
2  4 

The  series  then  is 

a-'  -  2a-'y-{-Sa-y -  4a-  y+5a-y,  &c. 

Here  the  law  of  the  progression  is  apparent ;  the  co-effi- 
cients increase  regularly  by  1,  and  their  signs  are  alternately 
positive  and  negative. 

481.  The  Binomial  Theorem  is  of  great  utility,  not  only 
in  raising  powers,  but  particularly  in  finding  the  roots  of  bino- 
mials. A  root  may  be  expressed  in  the  same  manner  as  a 
power,  except  that  the  exponent  is,  in  the  one  case  an  inte- 
ger, in  the  other  a  fraction.  (Art.  245.)  Thus  (a+6)"  may 
be  either  a  power  or  a  root.  It  is  a  power  if  n=2,  but  a  root 
if  n=i. 

482.  If  a  root  be  expanded  by  the  binomial  theorem,  the 
series  icill  never  terminate.  A  series  produced  in  this  way 
terminates,  only  when  the  index  of  the  leading  quantity  be- 
comes equal  to  0,  so  as  to  destroy  the  co-efficients  of  the  suc- 
ceeding terms.  (Art.  479.)  But  according  to  the  theorem, 
the  dilference  in  the  index,  between  one  term  and  the  next, 
is  always  a  unit ;  and  a  fraction,  though  it  may  change  from 
positive  to  negative,  cannot  become  exactly  equal  to  0,  by 
successive  subtractions  of  units.  Thus,  if  the  index  in  the 
first  term  be  J,  it  will  be, 

In  the  2d,     J  -  1  =  - 1,  In  the  4th  -  f  -  1  =  - 1, 

In  the  3d,  -  i  -  1  =  - 1,         In  the  5th  _  f  _  1  =  -  i,  &c. 

Ex.  What  is  the  square  root  of  {a-{-h)  1 
The  terms,  without  the  co-efficients,  are, 
aKa~h,a~  h\  a "  ^6^  a ~ h\  &c. 
32 


350  ALGEBRA. 

The  co-efficient  of  the  second  term  is  +1 

of  the  3d,  L^^i=  -i,  of  the  4tli,  ~^>^  ZI  =  Jr-,\, 

And  the  series  is  a'^-j-i^"^^-!^     "■+^^6^~''^^  &c. 

When  a  quantity  is  expanded  hy  the  Binomial  Theorem, 
the  law  of  the  series  will  frequently  be  more  apparent,  if  the 
factors,  by  which  the  co-efficients  are  fonned,  are  kept  dis- 
tinct. 

1.  Expand  into  a  series  {a'^-\-x)^. 
Substituting  b  for  a^,  we  have 

(6+  xy  =  b^+Ab~^x+Bb~^x-'-\-  Cb'^x'+Db~h\  &c 
j2  =  1,  (Art.  472.) 

2       2       2         4         2.4 


2.4        3  2.4  6     2.4.6 


2.4.6       4       2.4.6         8         2.4.6.8 

Restoring,  then,  the  value  of  6,  and  writing  -for  a~*,  we  have 

a 

^  ^         ^2a      2Aa'^2A.e,a'     2.4.6.8a' 

2.  Expand  into  a  series  (l-j-a:)^. 

Ans.  1+-  --^+-?^  -_M^,  &c. 
^2     2.4^2.4.6     2.4.6.8 


3.  Expand  V2,  or  (1  +  1)2. 


Ans.  1+1  -  --+—   -  _Ai-+_MlL_,  &c. 
2     2.4     2.4.6     2.4.6.8^2.4.6.8.10 


4.  Expand  (a+x)*,  or  a'^x  (l+-)'.     See  Art.  478. 

Ans.  a^x(l+— -    ""'    +     ^^'     -     ^-^^^ ,,  &c. 
\  ^2a     2.4a^^  2.4.6^^      2.4.6.8a^ 


INVOLUTION  OF  BINOMIALS.  'gSl 

5.  Expand  (a+6)%  or  aix(l+-)*. 

Ans.  a^  X    14- — -\ >  &c. 

6.  Expand  into  a  series  (a-b)  . 

Ans.alxfl-l-^-^:Z^---^^i^,&c.) 
V       4a     4.8a'     4.8.12a3     4.S.12.16a*         / 


1 


7.  Expand    (a-f.T)~^.     8.  Expand  (1- a:)  \ 
9.  Expand   (l+i-)  ~  t    10.  Expand  (a'+rr) ""  ^ 

483.  The  binomial  tlieorem  may  also  be  applied  to  quan- 
tities consisting"  of  more  than  two  terms.  By  substitution,  sev- 
eral terms  may  be  reduced  to  two,  and  when  the  compound 
expressions  are  restored,  such  of  them  as  have  exponents 
may  be  separately  expanded. 

Ex.  What  is  the  cube  of  a-\-b-\-c  1 

Substituting  h  for  (b-\-c,)  we  have  a-\-{b-{-c)=a-}-h. 

And  by  the  theorem,  (a+/i)'=a'+3a'/i+3a/i''+/i'. 

That  is,  restoring  the  value  of  h,         ^ 
{a+b+cy=a'+Sa'x{b+c)+Sax{b+cy^Hb+ey. 

The  two  last  terms  contain  powers  of  {b-\-c)  ;  but  these 
may  be  separately  involved. 

Promiscuous  Examples. 

1.  What  is  the  8th  power  of  {a-\-b)  1 

Ans.  a«+8a^fe+28a'6^4-56a^6^+70a*6*+56a^^  J- 
28aV+8a6'+6l 

2.  What  is  the  7th  power  of  (a  - 1)  ? 

3.  Expand  into  a  series ,  or  (1  -a)~^ 

1  -a 

Ans.  l+a+a'^+a'+a^.+a',  &c. 


352  ALGEBRA. 

4.  Expand  _^,  or  hxia-b)-'. 

A„s.  Ax(l+U^+^.  &c.)  o.-^+*^+^'+^f,  &c. 
\a     a''     a^     a^  /       «     a       a        a^ 

5.  Expand  into  a  series  (a^+6^)^. 

A            ,    b^      b'        b'      . 
Ans.  a4--  -— + ,  &c 

6.  Expand  into  a  series  {a-\-y)~\ 
Ans.  l-lM4-l?£-?2^_!+?_5C&c. 

7.  Expand  into  a  series  (c^-\-x^y. 

Ans.cx(l+^-?^+-M--l,&c.\ 
V  ^Sc'     3.6c^^3.6.9c^  / 

8.  Expand        ^„  or  ^^(c'^+a;'^)-^, 

c\       2c^^2Ac'    2.4.6c'^^2.4.6.8c«  ) 

9.  Find  the  5th  power  of  (a'+l/^) 

10.  Find  the  4th  power  of  (a+fe+n.) 

11.  Expand  {a'-xy.  12.  Expand  (l-y')^. 
13.  Expand  (a- a:) ^  14.    Expand /l(a^-^/')^ 


EVOLUTION.  253 

SECTION  XVllI. 
EVOLUTION  OF  COMPOUND  QUANTITIES. 

Art.  484.  THE  roots  of  compound  quantities  may  be  ex- 
tracted by  the  following  general  rule  : 

After  arranging  the  terms  according  to  the  powers  of  one 
of  the  letters,  so  that  the  highest  power  shall  stand  tirst,  the 
next  higliest  next,  &c. 

Take  the  root  of  the  first  term,  for  the  first  term  of  the  requir- 
ed root : 

Subtract  tJie  power  from  the  given  quantity,  and  divide  the 
first  term  of  the  remainder,  by  the  first  term  of  the  root  involved 
to  the  next  inferior  power,  and  multiplied  by  the  index  of  the 
given  power  ;\  the  quotient  toill  be  the  next  term  of  the  root. 

Subtract  the  power  of  the  terms  already  found  from  the  given 
quantity,  and  using  the  same  divisor,  proceed  as  before. 

This  rule  verifies  itself.  For  the  root,  whenever  a  new 
term  is  added  to  it,  is  involved,  for  the  purpose  of  subtract- 
ing its  power  from  the  given  quantity  :  and  when  the  power 
is  equal  to  this  quantity,  it  is  evident  the  true  root  is  found. 

Ex.  1.  Extiact  the  cube  root  of 

a'-\-Sa'  -  Sa'  -  1  la'+Grt'+12a  -  8(a'4-a  -  2. 
a®,  the  first  subtrahend. 


Sa"*)*     Sa%  &c.  the  first  remainder. 

a'-\-Sa'^Sa'-{-a\  the  2d  subtrahend. 


Sa^)*     *     -6a\  &c.  the  2d  remainder. 
'       a'-\-Sa' -  Sa'  -  lla^+6a'4-12a  -  8. 


fBy  the  given  poioer  is  mean*;  a  power  of  the  same  name  with  the  required 
root.  As  powers  and  roots  are  correlative,  any  quantity  is  the  square  of  its 
square  root,  the  cube  of  its  cube  root,  &c. 


254  ALGEBRA. 

Here  a^  the  cube  root  of  a",  is  taken  for  the  first  term  of 
the  required  root.  The  power  a^  is  subtracted  from  the  given 
quantity.  For  a  divisor,  the  first  term  of  the  root  is  squared, 
that  is,  raised  to  the  next  inferior  power,  and  multiphed  by 
3,  the  index  of  the  given  power. 

By  this,  the  first  term  of  the  remainder  3a^  &c.  is  divided, 
and  the  quotient  a  is  added  to  the  root.  Then  a^+a,  the 
part  of  the  root  now  found,  is  involved  to  the  cube,  for  the 
second  subtrahend,  which  is  subtracted  from  the  whole  of 
the  given  quantity.  The  first  term  of  the  remainder  -  M, 
&c.  is  divided  by  the  divisor  used  above,  and  the  quotient  -  2 
is  added  to  the  root.  Lastly  tlie  whole  root  is  involved  to 
the  cube,  and  the  power  is  found  to  be  exactly  eoual  to  the 
given  quantity. 

It  is  not  necessary  to  write  the  remainder  at  length,  as,  in 
dividing,  the  first  term  only  is  Avanted. 

2.  Extract  the  fourth  root  of 

a^+8a^+24a'+32a+16(a+2 


4aO*     8a^  &c. 


a4_|_8a3_j_24a''+32a+16. 

3.  What  is  the  5th  root  of 

a'j^5a'h+\0a^h''+\0d'h^+5ah'Jr^'  ?      Ans.  a+6. 

4.  What  is  the  cube  root  of 

a^-.Mh+\2ah''-mi  Ans.  a-26. 

6.  Wliat  is  the  square  root  of 

4a2- 12a6+96'4-16a;i- 246/i+16A^(2a-36+4/i 
4a* 


4a)*-12rt6,  &c. 
4a'^-l2a6+9fe' 


4a)*      *      *+       16  ah,  &c. 

4a'-  12a6+96^+16a/i-246/i+16/i'^. 


EVOLUTION.  255 

In  finding"  the  divisor  here,  the  term  2a  m  the  root  is  not 
involved,  because  the  power  next  below  the  square  is  the 
first  power. 

485.  But  the  square  root  is  more  commonly  extracted  by 
the  following  rule,  which  is  of  tJie  same  nature  as  that  which 
is  used  in  Arithmetic. 

After  arranging  the  terms  according  to  the  powers  of  one 
of  the  letters,  take  the  root  of  the  first  term,  for  the  first  term 
of  the  required  root,  and  subtract  the  pov/er  from  the  given 
quantity. 

Bring  down  two  other  terms  for  a  dividend.  Divide  by 
double  the  root  already  found,  and  add  the  quotient,  both  to 
the  root,  and  to  the  divisor.  Multiply  the  divisor  thus  in- 
creased, into  the  term  last  placed  in  the  root,  and  subtract 
the  product  from  the  dividend. 

Bring  down  two  or  three  additional  terms  and  proceed  as 
before. 

Ex.   1.  What  is  the  square  root  of 

a^,  the  first  subtrahend. 


2a+fe)*      2abJrb' 

Into  b=      2ab-\-b'^,  the  second  subtrahend. 


2a+26+c)  *     *    2ac+26c4-c2 

Into  c=  2ac-\-2bc-\-r,  the  third  subtrahend. 

Here  it  will  be  seen,  that  the  several  subtrahends  are  suc- 
cessively taken  from  the  given  quantity,  till  it  is  exhausted. 
If  then,  these  subtrahends  are  together  equal  to  the  square 
of  the  terms  placed  in  the  root,  the  root  is  truly  assigned  by 
the  rule. 

The  first  subtrahend  is  the  square  of  the  first  term  of  the 
root. 

The  second  subtrahend  is  the  product  of  the  second  term 
of  the  root,  into  itself,  and  into  twice  the  preceding  term. 

The  third  subtrahend  is  the  product  of  the  third  term 
of  the  root,  into  itself,  and  into  twice  the  sum  of  the  two  pre- 
ceding teriTis,  &c. 

That  is,  the  subtrahends  are  equal  to 

a2_|_(2a-j-6)  x6-j-(2a4-26+c)  xc,  &c. 
and  this  expression  is  equal  to  the  square  of  the  root. 


256  ALGEBRA. 

For  {a+by=a'+2ab-^b'=ia'-lr{2a+b)xh.   (Art.  120.) 
And  putting  h=a-\-b,  the  square  h'^=a'^-\-(2a-\-b)  X^- 
And  (a+6+c)^=:(/i+c)2  =  /i'^+(2/t+c)xc; 
that  is,  restoring  the  vahies  of  h  and  h^, 

(a^bJrcy=ar+{2a-\-b)  x6+(2rt+26+c)  xc 

In  the  same  manner,  it  may  be  proved,  that,  if  another 
term  be  added  to  the  root,  the  power  will  be  increased,  by 
the  product  of  that  term  into  itself,  and  into  twice  the  sum 
of  the  preceding  terms. 

The  demonstration  will  be  substantially  the  same,  if  some 
of  the  terms  be  negative. 

2.  What  is  the  square  root  of 

1  _  46+46^4-21/  -  46i/4-i/(l  -  2b+y 
1 


2  -  26)  *  -  46+462 
Into -26= -46+46- 


2  -  46+y)  *      *       2ij-  46y+i/^ 
Into       y=i  2ij  -  4by-\-'y'' 


3.  What  is  the  square  root  of 

a^  _  2a'+3rt'  -  2a'-^a''  1  Ans.  a'  -  cr~\-a. 

4.  What  is  the  square  root  of 

a^+4rt-6+46'^  -  4a'  -  86+4  *?  Ans.  rt''+26  -  2. 

486.  It  will  frequently  facilitate  the  extraction  of  roots, 
to  consider  the  index  as  composed  of  two  or  move  factors. 

Thus  a^=zrt^><^   (Art.  258.)    And  a'=:za'^^.    That  is, 

The  fourth  root  is  equal  to  the  square  root  of  the  square 
root ; 

The  sixth  root  is  equal  to  the  square  root  of  tlie  cube  root ; 

The  eighth  root  is  equal  to  the  square  root  of  the  fourth 
root,  &c. 

To  find  the  sixth  root,  therefore,  we  may  fust  extract  the 
cube  root,  and  then  the  square  root  of  this. 


ilVOLUTION.  257 

1 .  Find  the  square  root  of  x*  -  4a;^-f-6a;*  -  4a:+ 1 . 

2.  Find  the  cube  root  of  x'  -  6x'Jf-l5x'  -  20x^+1 5x* -  6x+l, 

3.  Find  the  square  root  of  4x^  - 4x^ -\-\ox^  -  6x-]-9. 

4.  Find  the  fourth  root  of 

16a'-96(ra:+216aV-216a.T'4-8lx*. 

5.  Fmd  the  5th  root  of  x'-^5x'-{-l0x^-\-10x''-{-5x-\-l. 

6.  Find  the  sixth  root  of 

a'  -  6a'b-^l5a'h'  -  20a'b'-\-loa'b'  -  6a¥+b\ 

ROOTS  OF  BINOMIAL  SURDS. 

486.  b.  It  is  sometimes  expedient  to  express  the  square 
root  of  a  quantity  of  tiie  form  aty\/b,  called  a  binomial  or  re- 
sidual surd,  by  the  sum  or  difference  of  two  other  surds.  A 
formula  for  this  purpose  may  be  derived  from  the  following 
propositions ; 

1.  The  square  root  of  a  whole  number  cannot  consist  of 
two  parts,  one  of  which  is  rational,  and  the  other  a  surd. 

If  it  be  possible,  let  .y/a—x-\-\/y,  in  which  the  part  x  is 
rational. 

Squaring  both  sides,  a=x'*-\-2x\/y-\-y 

And  reducing,  V!/— ~^>  "^  rational  quantity ; 

Zx 

which  is  contrary  to  the  supposition. 

2.  In  every  equation  of  the  form  x~\-\/y  =  a-\-\/b,  the  ra- 
tional parts  on  each  side  are  equal,  anjj  also  the  remaining 
parts. 

If  X  be  not  equal  to  a,  let  x=a1:z. 
Then  a±z-\-^y  =  a-{-\/b.  And  A,^/b=z-]-^yy  ; 

That  is,  y\/b  consists  of  two  parts,  one  of  which  is  rational, 
and  the  other  not ;  which,  according  to  the  preceding  propo- 
sition, is  impossible. 

In  the  same  manner  it  may  be  shewn,  that  in  the  equa- 
tion, a;  - /\/i/  =  a- \/6,  the  rational  parts  on  each  side  are 
equal,  and  also  the  remaining  parts. 


3.  If  \/a-\-A^b  =  x-\-A^y,  then  \/«  -  V^  =  ^- V!/- 

For,  by  squaring  the  first  equation,  we  have 

a-\-A^b  =  x'-{-2x.yyy-^y 

33 


258  ALGEBRA. 

And  by  the  last  proposition, 

\/b  =  2x/^y 


By  subtraction,  a-\/b  =  x'^-  2x\/y-\-y 
By  evolution,  Va  -  \/b  =  x  -  \/y. 

486.  c.  To  find,  now,  an  expression  for  the  square  root  of 
a  binomial  or  residual  surd. 

Let         \/a-\-\/b  =  x-{-\/y 

Then     \/a- \/b  =  x-A^y 
Squaring  both  sides  of  each,  we  have 

a  -  \/b  =  x^-  2xA^y-\-y 
Adding  the  two  last,  and  dividing,  a  =  x^-^y 

Multiplj'ing  the  two  first,  \/a^ -bznx^-y 

Adding  and  subtracting, 


a+ Va'  -  6  =  2a;^       Or  a—  V  2 


Ora— V 


a-\-\/ii^  -  b 


^a'-b  =  2y     AndV2/=>s/^— ^— 


Therefore,  as  \/a-\-\/b=x-\-/\/y,  and  /\/a-\/b=x-'\/yy 
Or,  svibstituting  d  for  ^a^  -  b, 

2.  v«-\/^=V4(^4-^)-V2(«-^)- 

Ex.  L  Find  the  square  root  of  3+2^2. 

Here  a=:3,  a'=9,  ^^=2^2,  ^=8,  a^-^  6=9  -  8=:L 

Therefore  v3+2v2=^^-|-^^  =  V^+L 


INFINITE  SERIES.  269 

2.  Find  the  square  root  of  11+6^2.  Ans.  3+^2. 

3.  Find  the  square  root  of  6-2^5.  Ans.  /y/5-  1. 

4.  Find  the  square  root  of  7+4^3.  Ans.  2-[->^3. 

5.  Find  tlie  square  root  of  7  -  2-y,/10.  Ans.  ^5  -  ^2. 

These  resuhs  may  be  verified,  in  each  instance,  by  multi- 
plying the  root  into  itself,  and  thus  re-producing  the  binomial 
from  which  it  is  derived. 


SECTION  XIX. 


INFINITE  SERIES, 


Art.  487.  IT  is  frequentty  the  case,  that,  in  attempting  to 
extract  the  root  of  a  quantity,  or  to  divide  one  quantity  by 
another,  we  find  it  impossible  to  assign  the  quotient  or  root 
with  exactness.  But,  by  continuing  the  operation,  one  term 
after  another  may  be  added,  so  as  to  bring  the  result  nearer 
and  nearer  to  the  value  required.  When  the  number  of 
terms  is  supposed  to  be  extended  beyond  any  determinate 
limits  the  expression  is  called  an  injimte  series.  The  quantity, 
however,  may  be  finite,  though  the  number  of  terms  be  un- 
limited. 

An  infinite  series  may  appear,  at  first  view,  much  less  sim- 
ple than  the  expression  from  which  it  is  derived.  But  the 
former  is,  frequently,  more  within  the  power  of  calculation 
than  the  latter.  Much  of  the  labor  and  ingenuity  of  mathe- 
maticians has,  accordingly,  been  employed  on  the  subject  of 
series.  If  it  were  necessary  to  find  each  of  the  terms  by  ac- 
tual calculation,  the  undertaking  would  be  hopeless.  But  a 
few  of  the  leading  terms  will,  generally,  be  sufificient  to  de- 
termine the  law  of  the  progression. 


SeO  ALGEBRA. 

488.  A  fraction  may  often  oe  expanded  into  an  infinite 
series,  by  dmding  the  numerator  by  tlie  denominator.  For  the 
value  of  a  fraction  is  equal  to  the  quotient  of  the  numerator 
divided  by  the  denominator.  (Art.  135.)  When  this  quotient 
cannot  be  expressed,  in  a  hmited  number  of  terms,  it  may  be 
represented  by  an  infinite  series. 

Ex.  To  reduce  the  fraction to  an  infinite  series, 

l-a 

divide  1  by  1  -  a,  according  to  the  rule  in  Art.  462. 

l_rt)l  (1  j^a-\-a''-\-a\  &c. 

l-a 


*     a\  &c. 


By  continuing  the  operation,  we  obtain  the  terms 

\-\-a-]-a^-\-a^-\-a^-\-a^-\-a^,  &c.  which  are  sufficient  to 
show  that  the  series,  after  the  first  term,  consists  of  the 
powers  of  «,  rising  regularly  one  above  another. 

That  the  series  may  converge^  that  is,  come  nearer  and 
nearer  to  the  exact  value  of  the  fraction,  it  is  necessary  that 
the  first  term  of  the  divisor  be  greater  than  the  second.  In 
the  example  just  given,  1  must  be  greater  than  a.  For  at 
each  step  of  the  division,  there  is  a  remainder  ;  and  the  quo- 
tient is  not  complete,  till  this  is  placed  over  the  divisor  and 
annexed.  Now  the  first  remainder  is  a,  the  second  a'^,  the 
third  a^,  &c.  If  a  then  is  greater  than  1,  the  remainder  con- 
tinually increases  ;  which  shoAVs,  that  the  farther  the  division 
is  carried,  the  greater  is  the  quantity,  either  positive  or  nega- 
tive, which  ought  to  be  added  to  the  quotient.  The  series 
is,  therefore,  diverging  instead  of  converging. 

But  if  a  be  less  than  1,  the  remainders,  a,  a^  a^  &c.  will 
continually  decrease.  For  powers  are  raised  by  multiplica- 
tion ;  and  if  the  multiplier  be  less  than  a  unit,  the  product 
will  be  less  than  the  multiplicand.  (Art.  90.)  If  a  be  taken 
equal  to  i,  then  by  Art.  223, 

0^=1  a'=\,  a'=h,  a'  =  ,\  &c. 


INFINITE  SERIES.  261 

and  we  have 
*   .=  l  =  2  =  l+i  +  i+i+^+,L+,V,  &c. 


I-i  i 

Here  the  two  first  terms  =l-f-i,  which  is  less  than  2,  by  J  ; 
the  three  first  =1-|-J,  less  than  2,  by  i  ; 

the  four  first  =  H-"8>  l^ss  than  2,  by  i  ; 

So  that  the  farther  the  series  is  carried,  the  nearer  it  ap- 
proaches to  the  vahie  of  the  given  fraction,  which  is  equal 
to  2. 

2.  If  be  expanded,  the  series  will  be  the  same  as  that 

l+a 

from  ,  except  that  the  terms  which  consist  of  the  odd 

l-a 

powers  of  a  will  be  negative. 

So  that  _L_=  1  -  a^ar  -  a^-\-a'  -  a'-^a',  &c. 

1-f-a 

3.  Reduce to  an  infinite  series. 

a-  b 

,  \ ,  (h  ,  bh  ,  b^    o 

a 


a 
Here  h  divided  by  a  gives  -  for  the  first  term  of  the  quo- 
tient.  (Art.  124.)  This  is  multiplied  into  a-b,  and  the  product 

is  h- —  ;  (Arts.  159,  158.)  which  subtracted  from  h  leaves 
a 

^.     This  divided  by  a  gives  ^  (Art.  163.)  for  the  second 
a  a 

term  of  the  quotient.     If  the  operation  be  continued  in  the 
same  manner,  we  shall  obtain  the  series, 

h  ,  bh  .  b^  .  ¥h  ,  b'h    p 

a     a^      a^      a^      a' 
in  which  the  exponents  of  b  and  of  a  increase  regularly  by  1. 


262  ALGEBRA. 

4.  Reduce    "^"-^  to  an  infinite  series. 

Ans.   14-2a+2a*+2a^+2aS  &c. 

489.  Another  method  of  forming  an  infinite  series  is,  by 
extracting  the  root  of  a  compound  surd. 


Ex.  1.  Reduce  Va-+6"^  to  an  infinite  series,  by  extracting 
the  square  root  according  to  the  rule  in  Art.  485. 

a^4-¥{  a-\- —  -  — o+ -c?  &'•• 


'O 


b^+J!- 

4a' 


^a     SaV  4a^ 


Here  a  the  root  of  the  first  terrn,  is  taken  for  the  first  term 
of  the  series  ;  and  the  power  a"^  is  subtracted  from  the  given 
quantity.      The  remainder  6^  is  divided  by  2a,  which  gives 

_- ,  for  the  second  term  of  the  root.     (Art.  124.)     The  divi- 

2a 

sor,  with  this  term  added  to  it,  is  then  multiphed  into  the 

term,  and  the  product  is  6"^+ — ,.      (Arts.  155,  159.)     This 

4rr 

subtracted  from  fe~  leaves  -  — ,   which  divided  by  2a  gives 

4a^ 

-  —,  for  the  third  term  of  the  root.     (Art.  163.)  &c. 

8a^  ^ 

b' 


2.   Va^-6^=a-^-i^-_%&c. 
2a    Sa^     16a' 


8.  V2=Vl+l  =  l+i-i+TV  &c. 
4.  A/TT^=l+-- -+—-—»  &c- 


INFINITE  SERIES.  263 

490.  A  binomial  which  has  a  negative  or  fractional  expo- 
nent, may  be  expanded  into  an  infinite  series  by  the  binomial 
theorem.  See  Arts.  480,  482,  and  the  examples  at  the  end 
of  Sec.  xvii. 


INDETERMINATE  CO-EFFICIENTS. 

490.  b.  A  fourth  method  of  expanding  an  algebraic  ex- 
pression, is  by  assuming  a  series,  with  indeterminate  co-effi- 
cients ;  and  afterwards  finding  the  value  of  these  co-efi[icients. 

If  the  series,  to  which  any  algebraic  expression  is  assumed 
to  be  equal,  be 

AJ^Bx-l-Cx^'+D^^+Ex',  &c. 

let  the  equation  be  reduced  to  the  form  in  which  one  of  the 
members  is  0.  (Art.  178.)  Then  if  such  values  be  assigned 
to.>5,  B,  C,  &c.  that  the  co-efiScients  of  the  several  powers 
of  X,  as  well  as  the  aggregate  of  the  terms  into  which  x  does 
not  enter,  shall  be  each  equal  to  0  ;  it  is  evident  that  the  wJiole 
will  be  equal  to  0,  and  that,  upon  this  condition,  the  equation 
is  correctly  stated. 

The  values  of  ^,  B,  C,  &:c.  are  detennined,  by  reducing 
the  equations  in  which  they  are  respectively  contained. 

Ex.  1.  Expand  into  a  series 


c-\-bx 


Assume   -l—=A-\-Bx.-{-Cx''-{-Dx'JrEx\  &c. 
c-{-bx 

Then  multiplying  by  the  denominator  c-\~bx,  and  trans- 
posing o,  we  have 

0={Ac'-a)-{-{Ab+Bc)x-\-(Bb-\-Cc)x^-\-(Cb-{-Dc)x^  &c. 

Here  it  is  evident,  that  if  {Ac -a),  (M-{-Bc),  {Bb-\-Cc), 
&c.  be  made  each  equal  to  0,  the  several  parts  of  the  second 
member  of  the  equation  will  vanish,  (Art.  112.)  and  the 
whole  will  be  equal  to  0,  as  it  ought  to  be,  according  to  the 
assumption  which  has  been  made. 


264  ALGEBRA. 

Reducing  the  following  equations, 


Jlc-a=0, 

we 

liav 

ew2=f!. 
c 

M+Bc=0, 

B=-^A 

C 

Bb-]-Cc=0, 

c=  -  ts, 

c 

Ch+Dc=0, 

c 

&c. 

&c. 

That  is,  each  of  the  co-efficients,  C,  D,  and  E,  is  equal  to 

the  preceding  one  multiplied  into  -  _.     We  liave  therefore 

c 

:_- X-\- -X^ X^-\- X\  &C. 


c-\-hx      c     c^        c^  c*  cP 

2.    Expand  into  a  series  ' 


d-]-hx-\~cx^ 


Assume       ^+^^     =A^Bx+Cx''-{-Dx\  &c. 

Then  multiplying  by  the  denominator  of  the  fraction,  and 
transposing  a-\-bx,  we  have  0={Ad- a)~\-{Bd-\-Ah-b)x 
+  {Cd+Bh-\-Ac)x''-\-{Dd+Ch-jrBc)x\  &c. 

Therefore  A=%  C=--B-  %A, 

a  d        d 

B=-^-\\,  D=--C-iB. 

d        d  d       d 

d^hx-^-cx"     d     \d       dj        \d    ^d    I 

3.  Expand  into  a  series       -r-^^    ^ 
^  l-x-x'' 

Ans.   l-\-Sx-\-4x''Jt-'7x'^Ux'J^\8j^-\-29x\  &c. 
In  which,  the  co-efficient  of  each  of  the  powers  of  x,  is  equal 
to  the  sum  of  the  co-efficients  of  the  tico  preceding  terms. 


4.  Expand  into  a  series 


INFINITE  SERIES.  2G5 

d 


b  —ax 


5.  Expand  into  a  series 


1  -  2a:  -  3a;* 
Ans.   l-^x-{-5x''-]-lSx'-\-4lx'-\-^2lx'-{-S65x\  &c. 

6.  Expand  into  a  series 

1  —  X  —  x'^-\-x^ 

Ans.   l-{-x+2x'-\-2x^Jr^x'-\-Sx'-\-4x'-{-4x\  &c. 

7.  Expand      ^     .  8.  Ex-paiid ^"^       . 

1-62'  ^         l-5a:+6a;» 

9.  Expand     ^+^^    .  10.   Expand  _1±^. 

SUiMMATION  OF  SERIES. 

^1.  Though  an  infinite  series  consists  of  an  unhmited 
number  of  terms,  yet,  in  many  cases,  it  is  not  ditiicult  to  find 
what  is  called  the  sum  of  the  temis ;  that  is,  a  quantity  which 
differs  less,  than  by  any  assignable  quantity,  from  the  value 
of  the  whole.  This  is  also  called  the  limit  of  the  series. — 
Thus  the  decimal  0.33333,  &c.  may  come  infinitely  near  to 
the  vulgar  fraction  i,  but  never  can  exceed  it,  nor,  indeed, 
exactly  equal  it.  See  Arts.  453,  4.  Therefore  i  is  the  limit 
of  0.33333,  &c.  that  is,  of  the  series 

I  0    I  "TD'Of   1    T  0  (TiT   I    I  0  b  OTT    i    T  0  0  0  0  0 5    *^^- 

If  the  number  of  terms  be  supposed  infinitely  great,  tJie 
difference  between  their  sum  and  ],  will  be  infinitely  small. 

492.  The  sum  of  an  infinite  series  whose  terms  decrease 
by  a  common  divisor,  may  be  found,  by  the  rule  for  the  sum 
of  a  series  in  geometrical  progression.     (Art.  442.)    Accord- 
ing to  this,  S=  ^^  ~  ^,  that  is,  the  smn  of  the  series  is  found 
r-  1 

by  multiplying  the  greatest  term  into  the  ratio,  subtracting 
the  least  term,  and  dividing  by  the  ratio  less  1.  But,  in  an 
infinite  series  decreasing,  the  least  term  is  infinitely  small. — 
It  may  be  neglected  therefore  as  of  no  comparative  value. 
(Art.  456.)     The  formula  will  then  become, 

r-1  r-1 

34 


2C<J  ALGEBRA. 

Ex.  1.  What  is  the  sum  of  the  infinite  series 

to    I     LOO    I     1000    I     L  0000    |     L  00000?   «^C.  f 

Here  the  fiist  tenn  is  -r^,  and  the  ratio  is  10 

Then  S=J1-  =l^l^=i-=i,  the  answer. 
r-l       10-1 

2.  What  is  the  sum  of  the  infinite  series 

Ans.  S=-^:f-=^^=2. 
r-l     2-1 

3.  What  is  the  sum  of  the  infinite  series 

l+i+i+2-^  +  8^1,  &c.  1  Ans.  f  =l+i. 

493.  There  are  certain  classes  of  infinite  series,  whose 
sums  may  be  found  by  subtraction. 

By  the  rules  for  the  reduction  and  subtraction  of  fractions, 
l_l_3-2_    1 

2  3~2x^3~2x3' 
l_l_4-3_    1 

3  4     3x4~3x4' 

l-l=^=_L,&c. 

4  5     4x5     4x5 

If  then  the  fractions  on  the  right  be  formed  into  a  series, 
they  will  be  equal  to  the  difference  of  two  series  formed  from 
the  fractions  on  the  left.  This  difference  is  easily  found  ; 
for  if  the  first  term  be  taken  away  from  one  of  these  two 
series,  it  will  be  equal  to  the  other. 

Suppose  we  have  to  find  the  sum  of  the  infinite  series 

-J 4- -\ ,  otc. 

2-3      3-4      4-5      5  6 

From  this,  let  another  be  derived,  by  removing  the  last 
factor  fiom  each  of  the  denominators ;  and  let  the  sum  of 
the  new  series  be  represented  by  S, 

That  is,  let         6^=1+1+1+1,  &c. 
2^3^4^5 

Then  ,S-1::=1+1+1+1,  &c.    . 

2     3^4^  5^6  ^ 


And  by  subtraction  l  =  J—+_-4 1 ,  &c. 

^  2     2-3^3-4^4-5^5-6 


INFINITE  SERIES.  2^*^ 

Here  the  new  series  is  made  one  side  of  an  equation,  and 
directly  under  it,  is  written  the  same  series,  after  the  first 
term  J  is  taken  away.  If  the  upper  one  is  equal  to  S,  it  is 
evident  that  the  lower  one  must  be  equal  to  S-  ^.  Then 
subtracting  the  terms  of  one  equation  from  those  of  the 
other,  (Ax.  2,)  we  have  the  sum  of  the  proposed  series 
equaltoj.     For  S  -  {S -i)  =  S  -  S-\-i=h 

2.  What  is  the  sum  of  the  infinite  series 

— 4- — 4- — -h — -T- — 5  &c. 

Here  a  new  series  may  be  formed,  as  before,  by  omitting" 
the  last  factor  in  each  denominator. 

Let  ,S=l+i+l+I+l,  &c. 

^2^3^4^5 

Then  S-  ^=i-fl +1+1-1-1,  &c. 

2     3^4^5^6^7^ 


'590         9         9         9 
And  by  subtraction  -=_r_+|_+_;_+_^_+i_,  &c. 

Or  2= _L+J_+J_+J_+J_,  &c. 
4     l-3^2-4^3-5^4-6^5-7 

In  repeating  the  new  series,  in  this  case,  it  is  necessary  to 
omit  the  two  first  terms,  which  are  l+i-=:f. 
3.  What  is  the  sum  of  the  infinite  series 

1.1.1,  1  c         rj 


2-4-6  '  4-6-8  '  6-8-10  '  8-1012 
Here  a  new  series  may  be  formed  by  omitting  the  last  fac- 
tor, and  retaining  the  two  first,  in  each  denominator.     And 
we  shall  find 

1        4,4,4.        4 


8     2-4-6^4-6-8'^6-810"^8-10-12' 


kc. 


Or  l  =  _L  +  _l_+_J_+_L_,&c. 
32     2-4-6^4-6-8^6-810^8-10-12 

4.  What  is  the  sum  of  the  infinite  series 

1  ,  1  ,  1  ,  1  o  «  .  1 

&c.  ?  Ans.  -. 


1-2-3  '  2-3-4     3-4-5  '  4-56  4 


268  ALGEBRA. 

493.  6.  Series  whose  sums  can  be  determined,  may  also 
be  found  by  the  following  method.  Assume  a  decreasing 
series,  containing  the  powers  of  a  variable  quantity  x,  whose 
sum  =  *S^.  Multiply  both  sides  of  the  equation,  by  a  com- 
pound factor,  in  which  x  and  some  constant  quantity  are  con- 
tained ;  and  give  to  x  such  a  value,  that  the  compound  fac- 
tor shall  be  equal  to  0.  If  one  or  more  of  the  first  terms  be 
then  transposed,  these  will  be  equal  to  the  sum  of  the  re- 
maining series. 

Ex.  1.  Let  S=\+-+-+-+-+-^  &c. 

Multiplying  both  sides  by  a:  -  1,  we  have 

^         ^  ^l-2^2-3^3-4^4-5^5-6 

If  we  make  .t=  1,  the  first  member  of  the  equation  becomes 
^X(l  -  1)  =  0.  (Art.  112.)  Then  transposing  -  1  from  the 
other  side,  we  have 

l'2^2-3^3-4^4-5^5-6 

2.  Let  S=l-^l-l.--\---\--,  &c.  as  before. 

^2^3^4^5 

Multiplying  by  o;^  -  1,  we  have, 

^  ^         .      2^1-3^2-4^3-5 

Making  x=l,  and  transposing  the  two  first  terms  of  the 
series,  we  have 

^2     2     l-3^2-4^3-5^4-6^5-7 

3.  Multiplying   ^^l+^+l'-f-,  &c.  by  2a;^-3a:+l, 

/i        o        4 

we  have 
^^  ^  ^  2^1-2-3^2'3-4^3-4-5 


And  if  X  be  put  equal  to  1 , 
3        5,6.7,8 


2     1-2-3  ■  2-3-4  '  3-4-5  '  4'5'6 


,  &c. 


INFINITE  SERIES.  269 

From  the  two  last  examples  it  will  be  seen,  that  different 
series  may  have  the  same  sum. 

RECURRING  SERIES. 

493.  c.  When  a  series  is  so  constituted,  that  a  certain 
nmnber  of  contiguous  terms,  taken  in  any  part  of  the  series, 
have  a  given  relation  to  the  term  immediately  succeeding, 
it  is  called  a  recurring  series ;  as  any  one  of  the  following 
tei-ms  may  be  found,  by  recurring  to  those  which  precede. 

Thus  in  the  series  \-\-^x+Ax'-\-lx^-\-Ux'-\-\^x\  &c. 

the  sum  of  the  co-efficients  of  any  two  contiguous  terms,  is 
equal  to  the  co-efficient  of  the  following  term.  If  the  series 
be  expressed  by 

^_|_j54-C+i)+£;,  &c. 

Then  .^=1,  the  first  term.     B=Si;  tlie  second, 
C=Bx-\-Jlx'^=4x\  the  third, 
D=Cx-{-Bx~=7x\  the  fourth,  &c. 

That  is,  each  of  the  terms,  after  the  second,  is  equal  to  the 
one  immediately  preceding  multiplied  by  x,  -j-  the  one  next 
preceding  multiplied  by  x\ 

In  the  series  lJf.2x-\-Sx'-\-4Qr-^5x'J^6x\  &c., 
each  term,  after  the  second,  is  equal  to  2x  multiplied  by  the 
term  immediately  preceding,  -x"^  multiplied  by  the  term 
next  preceding.    The  co-efficients  of  x  and  x"^,  that  is  -{-2  -  1, 
constitute  what  is  called  the  scale  of  relation. 

In  the  series  14-4a;-|-6a;^4-lla;3-f  28a;*+63r\  &c., 
any  three  contiguous  terms  have  a  constant  relation  to  the 
succeeding  term.  The  scale  of  relation  is  2  -  l-|-3  ;  so  that 
each  term,  after  the  third,  is  equal  to  2x  into  the  term  imme- 
diately preceding,  -  x^  into  the  term  next  preceding,  -j-Sa;^ 
into  the  third  preceding  term 

Let  any  recurring  series  be  expressed  by 

AJ^BJf-CJrD-irE+F,  &c. 

If  the  law  of  progression  depends  upon  two  contiguous 
terms  and  the  scale  of  relation  consists  of  two  parts,  m 
and  n. 


270  ALGEBRA. 

Then  C=Bmx-\-Anx\  tlie  third  term, 
D—Oinx-\-Bnx~,  the  fourth, 
E=Dmx-^Cnx\  the  fifth, 
&c.  &c. 

If  the  law  of  progression  depends  on  three  contiguous 
terms,  and  the  scale  of  relation  is  m-fn+r, 

Then  D=Cmx-\-Bnx'^-\-Arx^,  the  fourth  term, 
E=Dmx-\-Cnx'-\-Brx\  the  fifth, 
F=Emx-\-Dnx^-{-Crx^,  the  sixth, 
&c.  &c. 

If  the  law  of  progression  depends  on  more  than  three  terms, 
the  succeeding  terms  are  derived  from  them  in  a  similar 
manner. 

493.  d.  In  any  recurring  series,  the  scale  of  relation,  if  it 
consists  of  two  parts,  may  he  found,  by  reducing  the  equa- 
tions expressing  the  values  of  two  of  the  terms  ;  if  it  con- 
sists of  three  parts,  it  may  be  found  by  reducing  the  equations 
expressing  the  values  of  three  terms,  &c.  As  the  scale  of 
relation  is  the  same,  whatever  be  the  value  of  x  in  the  series, 
the  reduction  may  be  rendered  more  simple,  by  making  a:=  1. 

Taking  then  the  fourth  and  fifth  terms,  in  the  first  exam- 
ple above,  and  making  a;  =^1,  we  have 

^=  S'^+-^'^  I  to  find  the  values  of  m  and  n. 
E=Dm-{-Cn  ) 

These  reduced,  (Art.  339,)  give 
DC -BE  CE-DD 


m 


CC-BD  CC-BD 

ABODE        F 


In  the  series  |  ']_^3^+5^2_|_7^3^9:r^+lla;=,  &c. 

Making  x=z\,  we  have 

^^_7x5-3x9_,  n=^-^^=-\. 

5^-3x7  5^-3x7 

Therefore,  the  scale  of  relation  is  2  -  1 . 

To  know  whether  the  law  of  progression  depends  on  two, 
three,  or  more  terms  ;  we  may  first  make  trial  of  tw^o  terms  ; 
and  if  the  scale  of  relation  thus  found,  does  not  correspond 


INFINITE  SERIES.  271 

with  the  given  series,  we  may  try  three  or  more  terms.  Or 
if  we  begin  with  a  number  of  terms  greater  than  is  neces- 
sary, one  or  more  of  the  values  found  will  be  0,  and  the 
others  will  constitute  the  true  scale  of  relation. 

493.  e.  "When  the  scale  of  relation  of  a  decreasing  recur- 
ring series  is  known,  the  sum  of  the  terms  may  be  found. 

J        {A    B     C      D      E      F 

I  a^bx-\-cx''-irclx'-\-ex'-{-fa%  &c. 

be  a  recurring  series,  of  which  the  scale  of  relation  is  m-{-n. 
Then  ^=  the  first  term,         7?~  the  second, 
C=Bx^ix-{-*^Xnx^,  tfie  third, 
D=Cxmx-\-Bxnx\  the  fourth, 
JS=Dx»*a:4-Cx«ar^)  the  fifth, 
&c.  &c. 

Here  mx  is  multiplied  into  every  term,  except  the  first  and 
the  last ;  and  nx^  into  every  term  except  the  two  last.  If 
the  series  be  infinitely  extended,  the  last  terms  may  be  neg- 
lected, as  of  no  comparative  value,  (Art.  456,)  and  if  >Si=: 
the  sum  of  the  terms,  we  have 

S=.^+B-\-mxX{B+C-\-I>,  &c.)4-n.r'x(-^+5+C,  &c.) 
But  S-A=B+C-{-D,  &c.     And  S=Jl-\-B+C,  &c. 
Therefore  S=d-^B-\-^}xx{S-  A)-\-nx'xS. 
Reducing  this  equation,  we  have 
,^_Ji-{-B-Amx 
1  —  mx  —  nx^ 
Ex.  1.  What  is  the  sum  of  the  infinite  series 

lJ^6xJr^2x''+4Sx^-]-12Qx\  &c.  1 
The  scale  of  relation  will  be  found  to  be  l-j-6. 
Then.^r=l,         B=6x,       ?72=1,         n=6. 


The  series  therefore 


l-^5x 


1-x-Qx'' 
2.  What  is  the  sum  of  the  infinite  series 

f_|_3a;+4ar'-f-7a:3+lla;'+18a^'+29x^  &c.  1 

Ans.     '+'^\ 
l-x-x^ 


272  ALGEBRA. 

3.  What  is  the  sum  of  the  infinite  series 

l+x+5x'-\-lSa^-{.4\x'-{-n\x'^S65x\  &c.  1 

Ans.        *-^ 


1  -  2a;  -  3a,- 

4.  What  is  the  sum  of  the  infinite  series 

\+2x-\-Sx''-{-4x^-^5x\  &c.? 

Ans.  btl^^=  __1_ 
1- 2^:4-0;'      (l-x)' 

5.  What  is  the  sum  of  the  infinite  series 
l+Sx-i-5x''+7x'-\-9x'+Ux\  &c.? 

Ans.  — it^— 

6.  What  is  the  sum  of  the  infinite  series 

l+2a;+8a:^+28a;3+100a;S  &c.  ? 

Ans.         ^-^ 


1  -3a;-2a;2 

,f  .    ,1  .      (./2    J5     C      J9      J5      ii' 

W  m  tne  series  ^  a+bx+cx'-\-dx'+ex'-\-fx%  &c. 

the  scale  of  relation  consists  of  three  parts,  m-f-n-j-r, 

Then  ./Z=  the  first  term,      Bz=  the  second,     C=  the  third, 
D=Cx mx-{-B X nx'^-\-A x ?'a:^  the  fourth, 
E=Dxmx-{-Cxnx''-\~Bxrx\  the  fifth, 
jP=jEx«i^+-Dxw^''^+Cx^a;^  the  sixth, 
&c.  &c. 

Therefore 

S=A-]-B-]-C-}-mxx{C-^D-^E  &c.)-fn.T'^x 

(^_|_C+D  &c.)+ra;^x(^-f-e+C&c.)     That  is, 
S=^+BJrC+mxx(S-.^-B)-irnx"x{S-A)-\-rx'xS 
Reducing  this  equation,  we  have  • 

S=  ^^+^+<^-  (*^-\-B)mx-Mx'' 


INFINITE  SERIES.  373 

Ex.  1.  What  is  0ie  sum  of  the  infinite  series 

in  whicn  the  scale  of  relation  is  2  -  \-\-3  1 
Ans    ^+^^+^'-^^_{^+^r-'2x' 
1  -  2i:+a;-^  -  3a;^      '{l  -  xj'^-Sx'' 
2.  What  is  the  suni  of  the  infinite  series 
l+x-\-2x''-\-2x'-\-Sx'J^Sx'Jr4x'-{-4x\  &c. 
in  which  the  scale  of  relation  is  1-f-l  -  1? 

Ans.  ! 

I  —X  —  x^-i-x^ 

METHOD  OF  DIFFERENCES. 

493.  e.  In  the  Summation  of  Series,  the  object  of  inquiry 
is  not,  always,  to  determine  the  value  of  the  whole  when  in- 
finitely extended  ;  but  frequently,  to  find  the  sum  of  a  cer- 
tain number  of  tenns.  If  the  series  is  an  increasing  one,  the 
sum  of  all  the  terms  is  infinite.  But  the  value  of  a  limited 
number  of  terms  ma5^be  accurately  determhied.  And  it  is 
frequentl)"  the  case,  that  a  part  of  a  decreasing  series,  may 
be  more  easily  summed  than  the  whole.  A  moderate  num- 
ber of  terms  at  the  commencement  of  the  series,  if  it  conver- 
ges rapidly,  may  be  a  near  approximation  to  the  amount  of 
the  whole,  when  indefinitely  extended. 

One  of  the  methods  of  determining  the  value  of  a  hmited 
number  of  terms,  depends  on  finding  the  several  orders  of  dif^ 
ferences  belonging  to  the  series.  .The  differences  between 
the  terms  themselves,  are  called  the  first  order  of  differences; 
the  dilferences  of  these  differences,  the  second  order.  Sic.  In 
the  series, 

1,8,  27,64,  125,  &c. 

by  subtracting  each  term  from  the  next,  we  obtain  the  fiist 
order  of  differences 

7,  19,  37,61,  &c. 
and  taking  each  of  these  from  the  next,  we  have  the  second 
order, 

12,  18,  24,  &c. 
Proceeding  in  this  manner  with  the  series 
0,6,  c,  d,  e,f  &c. 
we  obtain  the  following  ranks  of  differences, 
35 


274  ALGEBRA 

1st.  Diff.  h-  a,  c-b,  d-c,  e-d,f-  e,  &c. 
2d.  Diff.  c-2b-{-a,  <i-.2c4-6,  e-2d^c,f-2e+d,  &:c. 
3d.  Diff.  d  -  3c-f-36  -  a,  e  -  3^+3c  -  b,  f-  Se-\-Sd  -  c,  &c. 
4th.  Diff.  e  -  4d-\-6c  -  4bJrCi,  f-  4e-\~6d-  4c-{-b   &c. 
5th.  Diff./-5e+10^-10c4-56-a,  &c. 
&:c.  »S>:c. 

In  these  expressions,  each  difference,  here  pointed  off  by 
commas,  though  a  compound  quantity,  is  called  a  term.  Thus 
the  first  term  in  the  first  rank  isb-a;  in  the  second,  c  -  2b-\-a ; 
in  the  third,  d-Sc-\-Sb-a;  &c.  The  first  terms,  in  the 
several  orders,  are  those  which  are  principally  emplo5^ed,  in 
investigating  and  applying  the  method  of  differences.  It  will 
be  seen,  that  in  the  preceding  scheme  of  the  successive  dif- 
ferences, the  co-efficients  of  the  first  term. 

In  the  second  rank,  are        1,     2,     1  ; 

In  the  third,  1,     3,     3,     1  ;  . 

In  the  fourth,  1,     4,     6,     4,     1  ; 

In  the  fifth,  1,     5,    10,    10,   5,     1  ; 

Which  are  the  same,  as  the  co-efficients  in  the  powers  of  bi- 
nomials. (Art.  471.)  Therefore,  the  co-efiicients  of  the  first 
term  in  the  nth  order  of  differences,  (Art.  472,)  are 

1  v.w- 1      ,  n-1     n~  2    9 

1,  n,  nx—^y  nX---X—^,  ^c- 

493.  /.  For  the  purpose  of  obtaining  a  general  expression 
for  an\^  term  of  the  series  a,  b,  c,  d,  Sic.  let  D\  D'\  D''\  D'''\ 
Sic.  represent  the  first  terms,  in  the  first,  second,  third,  fourth, 
&c.  orders  of  differences. 

Then  D^=b-  a, 

D''=zc-2b^a, 
D'''=d-Sc-^3b-a, 
D'''=e-  4c/+6c  -  46+a, 
&c.  &c. 

Transposing  and  reducing  these,  we  obtain  the  following 
expressions  for  the  terms  of  the  original  series,  a,  6,  c,  d,  &c. 

The  second  term  b  =  a-\-D\  ^ 

The  third,  c=aJ^2D'-\-D'\ 

The  fourth  d=: a+3jD^-f3D^^+D% 

The  fifth,  e  =  a-\.^D'J^^D''+^D"+jy"', 


INFINITE  SERIES.  275 

Here  the  co-efficients  observe  the  same  law,  as  in  the  pow^ 
ers  of  a  binomial;  with  this  difference,  that  the  co-efficients 
of  the  ?ith  term  of  the  series,  are  the  co-efficients  of  the 
(n-  l)th  power  of  a  binomial. 

Thus  the  co-efficients  of  the  fifth  term  are  1,  4,  6,  4,  1  ; 
which  are  the  same  as  the  co-efficients  of  the  fourth  power 
of  a  binomial.  Substituting,  then,  n  -  1  for  ?i,  in  the  formula 
for  the  co-efficients  of  an  involved  binomial,  (Art.  472,)  and 
applying  the  co-efficients  thus  obtained  to  D',  D'\  D['\  D'''\ 
&c.  as  in  the  preceding  equations,  we  have  the  following  gen- 
eral expression,  for  the  nth  term  of  the  series,  a,  b,  c,  d,  &c. 

The  nth  term 

=  a+(n-l)i?'+(n-l)'!rJi>"+„-l!^X^^^'",  &c. 

When  the  differences,  after  a  few  of  the  first  orders,  become 
0,  any  term  of  the  series  is  easily  found. 

Ex.  1.  What  is  the  nth  term  of  the  series  1,  3,  6,  10, 15,  21  1 

Proposed  series       1,  3,  6,  10,  15,  21,  &c. 

First  order  of  diff.    2,  3,  4,  5,  6,  &c. 

Second  do  1,   1,   1,  1,  &c. 

Third  do.  0,  0,  0, 

Herea=l,     D=2,     D''  =  h     D'''r-.0. 

fi  —  2 
Therefore  the  nth  term  r=l-|-(n- l)2+n- 1  _--. 

The  20th  term  =1+38+171=210.     The  50th  =  1 275. 

2.  What  is  the  20th  term  of  the  series  P,  2%  3^  4^  5\  &c.  t 

Proposed  series         1,     8,     27,     64,     125,  &c. 
First  order  of  diff        7,     19,     37,     61,  &c.  * 
Second  do.  12,     18,     24,  &c. 

""     Third  do.  6,       6,  &c. 

HereD'=7,       D''=12,       D'^^  =  6. 
Therefore  the  20th  term  =8000. 

3.  What  is  the  12th  term  of  the  series  2,  6,  12,  20,  30,  feci 

Ans.  156. 

4.  What  is  the  15th  term  of  the  series  l^  2\  3^  4\  5%  6\  &c.1 

Ans.  225. 


276  ALGEBRA. 

493.  g  To  obtain  an  expression  for  the  sum  of  any  number 
of  terms  of  a  series  a,  b,  c,  d,  &c.  let  one,  two,  three,  &c.  terms 
be  successively  added  together,  so  as  to  form  anew?  serieSy 

0,    a,    a-\-bj     a-\~b-\-Cj     a-\~b-\-c-\-d,  &c. 

Taking  the  differences  in  this,  we  have 

let    Diff.      a,         6,        c,       d,       e,        /,  &c. 
2d    Diff.    b-ttyC-by  d-c,  e-d,  f-  e,  &c. 
3d    Diff.  c  -  26+a,  d  -  2c-\-b,  e  -  2d-j-c,  f-  2e-\-dy  &c. 
4th  Diff.  d  -  3c+3fe  -  a,  e  -  3rf+3c  -  6,  /-  3e+3d  -  c,  &c. 
&c.  &c. 

Here  it  will  be  observed  that  the  second  rank  of  differences 
in  the  new  series,  is  /.he  same  as  the  first  rank  in  the  Griginal 
series  a,  6,  Cyd,  c,  &c.  and  generally,  that  the  (»i-|-l)th  rank 
in  the  new  series  is  the  same  as  the  nth  rank  in  the  original 
series.  If,  as  before,  D^=  the  first  term  of  the  first  differen- 
ces in  the  original  series,  and  d^=  the  first  term  of  the  first 
differences  in  the  new  series ; 

Then     d'=ay     d^'=zD\     d"'=D'^y     d^'''=D'''y  &c. 

Taking  now  the  formula  (Art.  493./.) 

Avhich  is  a  general  expression  for  the  nth  term  of  a  series  in 
which  the  first  term  is  a ;  applying  it  to  the  new  series,  in 
which  the  first  term  is  0,  and  substituting  n-\-\  for  ?i,  we  have 

[&c. 

Or  na+n!L-_ii>^+n--^  X^-?  D"+n^}z}.  x^-^-^X^=^i)-4- 

[&c. 
"Which  is  a  general  expression  for  the  (n-f-l)th  term  of  the 
series 

0,     a,     a-\~by     a-\~b-\-Cy     a-X.b-\-c-\-dy  &c. 

or  the  nth  terai  of  the  series 

a,     a-\-by     a-{-b-\-Cy     a-\~b-\-c-\-dy  &c. 
But  the  nth  term  of  the  latter  series,  is  evidently  the  sum 
of  n  terms   of  the  eericR,  a,   6,  c,  rf,   &c.      Therefore   tke 


INFINITE  SERIES.  277 

general  expression  for  the  sum  of  n  terms  of  a  series  of  which  a 
15  the  first  terrn^  is 

2  23  234 

[+&C. 

Ex.  1.  What  is  the  sum  of  n  terms  of  the  series  of  odd 
numbers,  t,  3,  5,  7,  9,  fccJ 

Series  proposed  !»  3,  5,  7,  9,  &c. 

First  order  of  diflf  2,  2,  2,  2,  &c. 
Second  do.  0,  0,  0, 

Here  «  =  !,  D  =2,         D"=Q. 

Therefore  the  sum  of  n  terms  =n-\-n^~    x^=n^- 

2 
That  is,  the  sum  of  the  terms  is  equal   to  ti\e  square  of  the 
immber  of  terms.     See  Art.  431. 

2.  What  is  the  sum  of  n  terms  o^  the  series 

P,  2\  S-,  4'-,  5\  &c.  1 
Herea=:l,         D'=S,         D''  =  2,         D'''=0. 

Therefore  n  terms  =i{2n^^Sn''-\-n)  :^in{nJ^\)  x  (2n+l). 
Thus  the  sum  of  20  terms  =2870. 

3.  What  is  the  sum  of  n  terms  of  the  series 

V,  2\  S\  4^  &C.1 
Herert=l,     I)'=7,     D'=\2,     D'''=6,     D''''=zO. 


Therefore  n  terms  =  J(n'4-2n'+n2)  =  (^wXM+l)l 
Thus  the  sum  of  50  terms  =1625625. 

4.  What  is  the  sum  of  n  terms  of  the  series     ' 
2,6,  12,  20,30,  SlcA 

Ans.  |n(n+l)x(n+2.) 

6.  What  is  the  sum  of  20  terms  of  the  series 
1,  3,  6,  10,  15,  &c.? 

6.  What  is  the  sum  of  12  terms  of  the  series 
l^  2^  3*,  4\  6\  &C.1  * 

♦See  Note  U. 


278  ALGEBRA. 


SECTION  XX. 


COMPOSITION  AND  RESOLUTION  OF  THE  HIGHER 
EQUATIONS. 


Art.  494.  EQUATIONS  of  any  degree  may  be  produced 
from  simple  equations,  by  multiplication.  The  manner  in 
which  they  are  compounded  will  be  best  understood,  by 
taking  them  in  that  state  in  which  they  are  all  brought  on 
one  side  by  transposition.  (Art.  178.)  It  will  also  be  neces- 
sary to  assign,  to  the  same  letter,  different  values,  in  the 
different  simple  equations. 

Suppose,  that  in  one  equation,  xz=2 
And,  that  in  another,  x=S 

By  transposition,  a^  -  2 = 0 

And  a; -3=0 


Multiplying  them  together,        a^  -  5a:-f-6=0 
Next,  suppose  a: -4=0 


And  multiplying,  a^  -  9x^-[-26x  -  24=0 

Again  suppose,  a;  -  5 = 0 

And  mult,  as  before,  a;*- 140,-^+713;'- 154a;+ 120=0,  &c. 
Collecting  together  the  products^  we  have 

(ar-2)(a:-3)  =ar^-5a:+6=0 

(a;- 2)  (a; -3)  (a: -4)      =a;^-9ar^-i-26a;- 24=0 

(a:-2)(a;-3)(a;-4)(a:-5)=a;*-14ar'+71a:'^-154ar4-120=0&c. 


EQUATIONS.  279 

That  is,  the  product 

of  two  smiple  equations,  is  a  quadratic  equation  ; 
of  three  simple  equations,  is  a  cubic  equation  ; 
of  four  simple  equations,  is  a  biquadratic,  or  an  equa- 
tion of  the  fourth  degree,  &c.   (Art.  300.) 

Or  a  cubic  equation  may  be  considered  as  the  product  of  a 
quadratic  and  a  simple  equation ;  a  biquadratic,  as  the 
product  of  two  quadratic  ;  or  of  a  cubic  and  a  simple  equa- 
tion, &c. 

495.  In  each  case,  the  exponent  of  the  unknown  quantity, 
in  the  first  term,  is  equal  to  tlie  degree  of  the  equation  ;  and, 
in  the  succeeding  terms,  it  decreases  regularly  by  1,  like  the 
exponent  of  the  leading  quantity  in  the  power  of  a  binomial. 
(Art.  468.) 

In  a  quadratic  equation,  the  exponents  are  2,  1. 

In  a  cubic  equation,  3,  2,  1. 

In  a  biquadratic,  4,  3,  2,  1,  &c. 

496.  The  number  of  terms,  is  greater  by  1,  than  the  degree 
of  the  equation,  or  the  number  of  simple  equations  from 
which  it  is  produced.  For  besides  the  terms  which  contain 
the  different  powers  of  the  unknown  quantity,  there  is  one 
which  consists  of  known  quantities  only.  The  equation  is 
here  supposed  to  be  complete.  But  if  there  are  in  the  partial 
products,  terms  which  balance  each  other,  these  may  disap- 
pear in  the  result.   (Art.  110.) 

497.  Each  of  the  values  of  the  unknown  quantity  is  cal- 
led a  root  of  the  equation. 

Thus,  in  the  example  above, 

The  roots  of  the  quadratic  equation  are  3,  2, 

of  the  cubic  equation  4,  3,  2, 

of  the  biquadratic  5,  4,  3,  2. 

The  term  root  is  not  to  be  understood  in  the  same  sense 
here,  as  in  the  preceding  sections.  The  root  of  an  equation 
is  not  a  quantity  which  multiplied  into  itself  will  produce  the 
equation.  It  is  one  of  the  values  of  the  unknown  quantity ; 
and  when  its  sign  is  changed  by  transposition,  it  is  a  term  in 
one  of  the  binomial  factors  which  enter  into  the  composition 
of  the  equation  of  which  it  is  a  root. 


280  ALGEBRA. 

The  value  of  the  unknown  letter  x^  in  the  equation,  is  a 
quantity  which  may  be  substituted  for  x,  without  affecting 
the  equality  of  the  members.  In  the  equations  which  we 
are  now  considering,  each  member  is  equal  to  0 ;  and  the 
first  is  the  product  of  several  factors.  This  product  will  con- 
tinue 10  be  equal  to  0,  as  long  as  any  one  of  its  factors  is  0. 
(Art.  112.)  If  then  in  the  equation 

(a;-2)x(^-3)x(^-4)-0r-5)=O, 
we  substitute  2  for  a:,  in  the  first  factor,  we  have 
Ox(a:-3)x(a'-4)-(a:-5)=zO. 

So,  if  we  substitute  3  for  x,  in  the  second  factor,  or  4  in 
the  third,  or  5  in  tlie  fourth,  the  w^hole  product  will  still  be  0. 
This  will  also  be  the  case,  when  the  product  is  formed  by  an 
actual  multiplication  of  the  several  factors  into  each  other. 

Thus,  as  a:3-92^+26a;-24zr:0  ;    (Art.  494. 
So     23  -  9  X  2^+26  X  2  -  24  =  0, 
And3'-9x3^+26x3-24:=0,  &c. 

Either  of  these  values  of  x,  therefore,  will  satisfy  the  con- 
ditions of  the  equation. 

498.  The  number  of  roots,  then,  which  belong  to  an  equa- 
tion, is  equal  to  the  degree  of  the  equation. 

Thus,  a  quadratic  equation  has  two  roots  ; 
a  cubic  equation,  three ; 
a  biquadratic, /owr,  &c. 

Some  of  these  roots,  however,  may  be  imaginary.  For  an 
imaginary  expression  may  be  one  of  the  factors  from  which 
the  equation  is  derived. 

499.  The  resolution  of  eqviations,  which  consists  in  finding 
their  roots,  cannot  be  well  understood,  without  bringing  into 
view  a  number  of  principles,  derived  from  the  manner  in 
which  the  equations  are  compounded.  The  laws  by  which 
the  co-efficients  are  governed,  may  be  seen,  from  the  following 
view  of  the  multiplication  of  the  factors 

X  -a,  x-b,  x-c,  X-  d, 

each  of  which  is  supposed  equal  to  0. 

The  several  co-efficients  of  the  same  power  of  x,  are  pla- 
ced under  each  other. 


EQUATIONS.  281 

Thus,  -ax  -  bx  is  written  ~  ?  (  ^  ;    aiid  the  other  co-effi- 
cients in  the  same  manner. 

The  product,  then 
Of    {x-a)  =  0 
Into  (a;- 6)  =0 


-^^? 


Is  a:^_i  (  x-\-ab=:0,  a  quadratic  equation. 
This  into  x-  c  =  0 


-a)     -{-ah 
Is  ar*  -  6  >  x^-\-(LC  }  x-  abc=0,  a  cubic  equation. 
-c)     -\-bc 
Tliis  into  x  -dz=0. 
ah 

-  ahc 


is.'-H 


-\-ac 

-  c  (  '^  -\-bc 
-dj       J^bd 
-{.cd 


>  X  _  „^j  ^  x-\-abcd=:0,  a  biquadratic. 


aci 
bed 


&c. 


500.  By  attending  to  these  equations,  it  will  be  seen  that, 

In  the^rs^  term  of  each,  the  co-efficient  of  a;  is  1  : 

In  the  second  term,  the  co-efficient  is  the  sum  of  all  the 

roots  of  the  equation,  Avith  contrary  signs.     Thus  the  roots 

of  the  quadratic,  equation  are  a  and  h,  and  the  co-efficients, 

in  the  second  term,  are  -  a  and  -  b. 

In  the  third  term,  the  co-efficient  of  x,  is  the  sum  of  all 
the  products  which  can  be  made,  by  multiplying  together 
any  two  of  the  roots.  Thus,  in  the  cubic  equation,  as  the 
roots  are  a,  b,  and  c,  the  co-efficients,  in  the  third  term,  are 
ah,  ac,  be. 

In  the  fourth  term  the  co-efficient  of  x  is  the  sum  of  all 
the  products  which  can  be  made,  by  multiplying  together 
any  three  of  the  roots  after  their  signs  are  changed.  Thus 
the  roots  of  the  biquadratic  equation  are  a,  6,  c,  and  d,  and 
the  co-efficients  in  the  fourth  term  are  -  abc^  -  abd,  -  acd, 
-bed. 

The  last  term  is  the  product  formed  from  all  the  roots  of 
the  equation  after  the  signs  are  changed. 

36 


282  ALGECRV. 

Iti  the  cubic  equation,  it  is-a)^-6x  -c=  -  abe. 
In  the  biquadratic,  -ax  -^X  -cX  -d—-{-abcd,  &c. 

501.  In  tlie  preceding  examples,  the  roots  are  all  positive. 
The  signs  are  clianged  by  transposition,  and  when  the  seve- 
ral factors  are  multiplied  together,  the  terms  in  the  product, 
as  in  the  power  of  a  residual  quantity,  (Art.  476,)  are  alter- 
nately positive  and  negative.  But  if  the  roots  are  all  nega- 
tive, they  become  positive  by  transposition,  and  all  the  terms 
in  the  product  must  be  positive.  Thus  if  the  several  values 
of  X  are  -a,  -  b,  -  c,  -  d,  then 

x-\-a  =  0,  x-[-b  —  0,x-\-c=LO,  x-{-d=0 ; 

and  by  multiplying  tliese  togetlier,  we  shall  obtain  the  same 
equations  as  before,  except  that  the  signs  of  all  the  terms 
will  be  positive.  In  other  cases,  some  of  the  roots  may  be 
positive,  and  some  of  tliem  negative. 

502.  As  equations  are  raised,  from  a  lower  degree  to  a 
hitler,  by  multiplication,  so  they  may  be  depressed,  from  a 
higher  degree  to  a  lewder,  by  division.  The  product  of  (x  -  a) 
into  (x  -  6)  is  a  quadratic  equation  ;  this  into  {x  -  c)  is  a 
cubic  equation  ;  and  this  into  (x-d)  is  a  biquadratic.  (Art. 
494.)  If  we  reverse  this  process,  and  divide  the  biquadratic 
by  (x-d),  the  quotient,  it  is  evident,  will  be  a  cubic  equa- 
tion ;  and  if  w^e  divide  this  by  (x  -  c)  the  quotient  will  be 
quadratic,  »&:c.  The  divisor  is  one  of  the  factors  from  w^hich 
the  equation  is  produced;  that  is,  it  is  a  binomial  consisting 
of  X  and  one  of  the  roots  with  its  sign  changed.  When, 
therefore,  we  have  found  either  of  the  roots,  w^e  may  divide 
by  this,  connected  with  the  unknown  quantity,  which  will 
reduce  the  equation  to  the  next  inferior  degree. 

RESOLUTION  OF  EQUATIONS. 

503.  Various  methods  have  been  devised  for  the  resolution 
of  the  higher  equations  ;  but  many  of  them  are  intricate  and 
tedious,  and  others  are  applicable  to  particular  cases  only. 
The  roots  of  numerical  equations  may  be  found,  however, 
with  sufficient  exactness  by  successive  approximations.  From 
the  laws  of  the  co-efficients,  as  stated  in  Art.  500,  a  general 
estimate  may  be  formed  of  the  values  of  the  roots.  They 
must  be  such,  that,  wiien  their  signs  are  chaiiged,  their 
product  shall   be  equal  to  the  last  term  of  the  equation,  and 


EQUATIONS.  283 

their  sum  equal  to  the  co-efficient  of  the  second  term.  A  trial 
may  then  be  made,  by  substituting,  in  the  place  of  the  un- 
known letter,  its  supposed  value.  If  this  proves  to  be  too 
small  or  too  great,  it  may  be  increased  or  diminished,  and 
the  trials  repeated,  till  one  is  found  which  Avill  nearly  satisfy 
the  conditions  of  the  equations.  After  we  have  discovered  or 
assumed  two  approximate  values,  and  calculated  the  errors 
which  result  from  them,  we  may  obtain  a  more  exact  cor- 
rection of  the  root,  by  the  following  propGrtion. 

As  the  difference  of  the  errors,  to  the  difference  of  the  assumed 
numbers ; 

So  is  the  least  error,  to  the  correction  required,  in  the  corres- 
Xfonding  assumed  number. 

This  is  founded  on  the  supposition,  that  the  errors  in  the 
esults  are  proportioned  to  the  errors  in  the  assumed  numbers. 

Let  JV  and  n  be  the  assumed  numbers  ; 

>S  and  s,        the  errors  of  these  numbers  ; 

R  and  r,        the  errors  in  the  results. 

Then  by  the  supposition  R:  r  : :  S  :  $ 

And  subt.  the  consequents  (Art.  3S9.)  R-r  :  S -s  :  :r  :  s. 

But  the  difference  of  the  assumed  numbers  is  the  same, 
as  the  difference  of  their  errors.  If  for  instance,  the  true 
number  is  10,  and  the  assumed  numbers  12  and  15,  the  er- 
rors are  2  and  5 ;  and  the  difference  between  2  and  5  is  the 
same  as  between  12  and  15.  Substituting,  then,  JV-n  for 
S -s,  we  have  R  -r  :  JST-n:  :  r  :  s,  which  is  the  proportion 
stated  above. 

The  term  difference  is  to  be  underslood  here,  as  it  is  com- 
monly used  in  algebra,  to  express  the  result  of  subtraction 
according  to  the  general  rule.  (Art.  82.)  In  this  sense,  the 
difference  of  two  numbers,  one  of  which  is  positive  and  the 
other  negative,  is  the  same  as  their  5W7)i  would  be,  if  their 
signs  were  alike.      (Art.  85.) 

The  supposition  which  is  made  the  foundation  of  the  rule 
for  finding  the  true  value  of  the  root  of  an  equation,  is  not 
strictly  correct.  The  errors  in  the  results  are  not  exactly 
proportioned  to  the  errors  in  the  assumed  numbers.  But 
as  a  greater  error  in  the  assumed  number,  will  generally  lead 
to  a  gieater  error  in  the  result,  than  a  less  one,  the  rule  will 
answer  the  purpose  of  approximation.     If  the  value  which  is 


284  ALGEBRA. 

first  found,  is  not  sufficiently  correct,  this  maybe  taken  as  one 
of  the  numbers  for  a  second  trial ;  and  the  process  may  be 
repeated  till  the  error  is  diminished  as  much  as  is  required. 
There  will  generally  be  an  advantage  in  assuming  two  num- 
bers whose  difference  is  .1,  or  .01,  or  .001,  &c. 

Ex.  1.  Find  the  value  of  x,  in  the  cubic  equation, 

a:3_8a;^+17a;-10=0. 

Here  as  the  signs  of  the  terms  are  alternately  positive  and 
negative,  the  roots  must  be  all  positive  ;  (Art.  501.)  their 
product  must  be  10  and  their  sum  8. 

Let  it  be  supposed  that  one  of  them  is  51  or  5-2.  Then, 
substituting  these  numbers  for  x,  in  the  given  equation,  we 
have, 

Bytlielstsuppos'n,(5-l)'-8x(5-l)2+17x(5-l)-10=l-271. 
By  the  second  (3'2)^ -8x  (5'2)2+17x (5-2)  -  10  =  2-688. 
That  is,  By  the  first  supposition.    By  the  second  supposition, 

The  1st  term,  x^=      132-651  140608 

The  2d  -8a;2=  -208-08  -216-32 
The  3d                   Ux=       86.7  88-4 

The  4th  -10=-    10.  -    10- 


Sums  or  errors,  +1-271  +2-688 

Subtracting  one  from  the  other,  1*271 


Their  difference  is  1-417 

Then  stating  the  proportion 
1-4:0-1::  1-27  :  0-09,  the  correction  to  be  sub- 
tracted from  the  first  assumed  number  5*1  :  The  remainder 
IS  5*01,  which  is  a  near  value  of  x. 

To  correct  this  farther,  assume  ar=5-01,  or  5-02. 
By  the  first  supposition.  By  the  second  supposition 

The  1st  term        x'=     125-751  126-506 

-201-6 

85-34 
^10. 
^ 

Errors  +    0121  +  0-246 

0-121 


The  2d 
The  3d 
The  4th 

-8a;^= -200-8 
\7x  =       85-17 
-10  =  -    10- 

Difference  0125 


EQUATIONS.  285 

Then  Q\25  i  0-01  :  :  0-121  :  001,  the  correction.  This 
subtracted  from  5*01,  leaves  5  for  the  value  of  x;  which  will 
be  found,  on  trial,  to  satisfy  the  conditions  of  the  equation. 

For  53-8x5^+17x5-10=0. 

We  have  thus  obtained  one  of  the  three  roots.  To  find 
the  other  two,  let  the  equation  be  divided  by  a;  -  5,  accordnig 
to  Art  462,  and  it  will  be  depressed  to  the  next  inferior  de- 
gree.  (Art.  502.) 

X  -  5)x^  -  8:k-+17.t  -  10(a;-  -  3a'+2  =  0. 
Here,  the  equation  becomes  quadratic. 
By  transposition,  x'^-Sx=  -  2. 

Completing  the  square,  (Art.  305.)  x'^-Sx-\-i=i-2  =  l, 
Extract,  and  transp.  (Art.  303,)     x  =  S±A/^=i±i. 
The  first  of  these  varies  of  x,  is  2,  and  the  other  1. 

We  have  now  found  the  three  roots  of  the  proposed  equa- 
tion. When  their  signs  are  changed,  their  sum  is  -  8,  the 
co-efidcient  of  the  second  term,  and  their  product  -  10,  the 
last  term. 

2.  What  are  the  roots  of  the  equation 

a;3_8a:-+4a;-|-48  =  0'?  Ans.  -2,+4,+6. 

3.  What  are  the  roots  of  the  equation 

a;3_  lGx^j^65x- 50  =  01  Ans.  1,  5,  10. 

4.  What  are  the  roots  of  the  equation 

aP-\-2x'  -  33:r  =  90  ?  Ans.  6,-5,-  3. 

5.  AVhat  is  a  near  value  of  one  of  the  roots  of  the  equation 

x^-{-9x^-]-.4x=801 

6.  What  is  a  near  value  of  one  of  the  roots  of  the  equation 

x^-{-x^-^x=lOOl 

503.  b.  Another  method  of  approximating  to  the  roots  of 
numerical  equations,  is  that  of  Newton,  by  successive  substi- 
tutions. 

Let  r  be  put  for  a  number  found  by  trial  to  be  nearly  equal 
to  the  root  required,  and  let  z  denote  the  difference  between  r 
and  the  true  root  x.  Then  in  the  given  equation,  substitute 
rtz  for  X,  and  reject  the  terms  which  contain  the  powers  of  z. 


286  ALGEBRA. 

This  will  reduce  the  equation  to  a  simple  one.  And  if  z 
be  less  than  a  unit,  its  powers  will  be  still  less,  and  therefore 
the  error  occasioned  bythe  rejection  of  the  terms  in  which 
they  are  contained,  will  be  comparatively  small.  If  the 
value  of  r,  as  found  by  the  reduction  of  the  new  equation, 
be  added  to  or  subtracted  from  r,  according  as  the  latter  is 
fomid  by  trial  be  too  great  or  too  small,  the  assumed  root  will 
be  once  corrected. 

By  repeating  the  process,  and  substituting  the  corrected 
value  of  r,  for  its  assumed  value,  we  may  come  nearer  and 
nearer  to  the  root  required. 

Ex.  1.  Find  one  of  the  values  of  cc,  in  the  equation 

% 
het  r-z=zx. 

C  x^^=(r  —  zy=z  r^  —  Sr^z-\-  ^z"^  —  z'^  ") 

Then  )  -  16V=:  -  U{r-zy=  -  16r'-|-32r^- I652  V  =50. 

(      65x  =     65(r  — £)  =     65r  ~65z  ) 

Rejecting  the  terms  which  contain  z"  and  z^,  we  have 
r3  _  ]  6r''-{-6or  -  Sr''z-\-S2rz  -  65z=50. 

•  .This  reduced  gives  , 

^~    -3r  +32r-65' 
If  r  be  assumed  =11,  then  z  —  —  =  0-8  nearly. 

anda:=r~2r  nearly  =11 -0-8  =  10-2. 

To  obtain  a  nearer  approximation  to  the  root,  let  the  cor- 
rected value  of  10-2  be  now  substituted  for  r,  in  the  preceding 
equation,  instead  of  the  assumed  value  11,  and  we  shall  have 

-=•188  rr=r-z=10-012. 

For  a  fMrc/ approximation,  let  r=10  012,  and  we  have 
2  = -012  a;=r-2:=10. 

2.  What  is  a  near  value  of  one  of  the  roots  of  the  eqitation 
x^-\-\0x''-\-5x=2600l  Ans.  11-0067. 

3.  What  are  the  roots  of  the  equation 

x'-{-2x''-Ux=\2l 


EQUATIONS.  287 

4.  What  are  the  roots  of  the  equation 

a:*4-4:r3_7a:'-34a'  =  24'? 

503. c.  An  equation  of-  the  »^th  degree  consists  of  af*,  the 
several  inferior  powers  of  x  with  their  co-efficients,  and  one 
term  in  wliich  x  is  not  contained.  If  A,  B,  C,  .  .  ,  ,  T,  be 
put  for   the  several  co-efficients,  and   U  for  the  last  term, 

then  a:'"+.^a;-"-i-f-^u:-"-'-j-Cx"'-3 4-Ta;4-C/=0, 

will  be  a  general  expression  for  an  equation  of  any  degree. 

If  a,  b,  c,  &c.  be  7'oots  of  any  equation,  tliat  is,  such  quan- 
tities as  may  be  substituted  for  x ;  (Art.  497.)  it  maybe 
shown,  without  reference  to  the  method  of  producing  the 
equation  by  multiplication,  that  the  first  member  h  exactly 
divisible  by  x  -  a,  x-b,  x-c,  &c. 

For  by  substituting  a  for  x,  we  have 

Ana  transposing  terms, 

U=  -  a"^  -Jla"-'  -  Ba""-^  -  Ccr-\ -  Ta. 

Substituting  this  value  for  U,  in  the  original  equation, 

x'^-\-Jlx'^-'-\-Bx'^-''-\-Cx'--^ -\-Tx  >  __^ 

^-ar-Aar-'-BaT-^-  Ca"'-^ -  Ta]  -^' 

•<* 
Or,  uniting  the  correspondin^g  terfns, 
(a;-  _  ^-)_|_(,^a;— 1  -  Aa'^-')-i^(Bx^-'  -  Bar-'')-\- 
ICx"^-^  -  Car-^)  ....  -^T{x-a)^0. 

In  this  expression,  each  of  the  quantities  [x"^~ct), 
{AaT-^  -  Aa"^-^),  &c.  is  divisible  by  rr-  a ;  (Art.  466.)  there- 
fore the  whole  is  divi*sit>le  b)^  x  -  a. 

In  the  same  manner  it  may  be  shown,  that  the  equation  is 
divisible  by  a;  -  6,  x-  c,  &c.  ^ 

50S.d.  The  quotient  produced  by  dividing  the  original 
equation  by  a;  -  a,  is  evidently  equal  to  the  aggregate  of  the 
particular  quotients  arising  from  the  division  of  the  several 
quantities  (x^-a""),  {x""-' -a""-'),  &c. 

The  quotient  of  {x"^-  a"')-^{x-  a),  (Art.  466)  is 

The  quotient  of  A  {af~^  -a"~')-f-(a:-rt)  is 
&c.  &c. 


288  ALGEBRA. 

Collecting  these  particular  quotients  together,  and  placing 
under  each  other  the  co-efficients  of  the  same  power  of  x,  we 
have  the  following  expression  for  the  quotient  of 

x'"^Jlx—'-\-Bx'"-^-i-Cx'^-'  ....  -^Tx+U 

divided  by  x  -  a. 

-\-A  r        +^ct\  X-'  +.^a^  (  ^_,         +  At-» 

The  quotient  of  the  same  equation  divided  by  x-b,is 
+J1  I  -^       +Ab  [  .r"-'+^6^  (  +^i- 

II.  +C     )  +01"-' 


+  T. 
The  quotient  from  dividing  by  a;  -  c,  is 

III.  -{-C    )  -\-Cc- 


+  T. 


In  the  same  manner  may  be  found  the  quotients  produced 
by  introducing  successively  into  the  di^sor  the  several  roots 
of  the  equation  ;  which  are  equal  in  number  to  m. 

503. e.  From  the  known  relations  between  the  roots  and 
the  co-efficients  of  equations,  as  stated  in  Art.  500,  Newton 
has  derived  a  method  of  determining  the  co-efficients,  from 
the  sum  of  the  roots,  the  sum  of  their  squares,  the  sum  of 
their  cubes,  &c.,  though  the  roots  themselves  are  unknown  ; 
and  on  the  other  hand  of  determining  from  the  co-efficients, 
the  sum  of  the  roots,  the  sum  of  their  s-quares,  the  sum  of 
their  cubes,  &c.  For  this  purpose,  the  following  plan  of  no- 
tation is  adopted.  Si  is  put  for  the  sum  of  the  roots,  S^  for 
the  sum  of  their  squares,  S3  for  the  sum  of  their  cubes,  tj-c. 
If  the  roots  are  a,  6,  c,  rf,  .  .  .  /,  then 


EQUATIONS.  289 

Si=a-\-b-}-cJf-d  ..  .  -}-Z 

S^  =  a'^-J^b"' -{•€"' Jrd'" .  . .  +/" 

&C.  &€. 

By  means  of  this  notation,  we  obtain  the  following  expres- 
sion for  the  sum  of  all  the  quotients  marked  I,  II,  III,  &c. 
(Art.  oOS.d.)  and  continued  till  their  number  is  equal  to  m. 


-2+       ^2] 

)                   +        *^3 

-^^s,\ 

>a:'"-^-t-.5^^2 

-{-mB  : 

)           -{.BS\ 

r. 

+mC 

-fmr. 


In  the  original  equation, 

x'"-\-Jlx'"-'-\-Bx"'-^-\-Cx'^-' .  .  .  J^Tx-}-U=0, 

the  co-efficients,  .^,  j&,  C,  &c.  have  determinate  relations  to 
the  sum  and  products  of  the  roots,  a,  b,  c,  &c.  (Art.  500.) 
But  the  quotient  marked  I,  (Art.  503.  d.)  produced  by  divid- 
ing by  x-a,  is  the  first  member  of  an  equation  of  the  next 
inferior  degree,  (Art.  502.)  from  which  the  root  a  is  excluded. 
So  b  is  excluded  from  the  quotient  II,  c  from  the  quotient  III, 
&c.  In  the  expression  above  marked  F,  which  is  the  sum 
of  m  quotients,  the  co-efficient  of  x  in  the  second  term  is 
Si  -\-m^.  But  ^i,  which  is  the  co-efficient  of  x  in  the  second 
term  of  the  original  equation,  is  equal  to  the  sum  of  the 
roots  a,  b,  c,  &c.  with  contrary  signs;  (Art.  500.)  that  is 
Si  =  -^.     Therefore, 

Si-^mA={m-l).^. 

In  the  third  term  of  the  original  equation,  B  the  co-effi- 
cient of  x,  is  equal  to  the  sum  of  all  the  products  which  can 
be  made  by  multiplying  together  any  two  of  the  roots.  (Art. 
500.)  But  each  of  these  products  will  be  excluded  from 
two  of  the  quotients,  I,  II,  III,  &c.  For  instance,  ab  will  not 
be  found  in  the  first,  from  wiiich  a  is  excluded,  nor  in  the 
second,  from  which  b  is  excluded.  Therefore  in  the  expres- 
sion F,  the  co-efficient  of  x  in  the  third  term  is  equal  to 

37 


290  ALGEBRA. 

mB  -  Zab  -Zac-  2ad,  &c.     But  -  2afe,  -  2ac,  -  2ad,  &c.  =  - 
2^.     So  that 

S^-\.AS,+mB=  {m-2)B. 

In  the  fourth  term  of  the  original  equation,  C  the  co-eflfi- 
cient  of  x,  is  equal  to  the  sum  of  all  the  products  which  can 
be  made  by  multiplying  together  any  three  of  the  roots,  after 
their  signs  are  changed.  But  each  of  these  products  will  be 
excluded  from  three  of  the  quotients,  I,  II,  III,  &c.  So  that, 
in  the  expression  F,  the  co-efficient  of  x  in  the  fourth  term, 
i^ equal  to  mC-Sabc  -  Sabd,  &c.     That  is, 

S^-\-AS,-]-BS,-{-mC=  (m  -  3)  C. 

In  the  same  manner,  the  values  of  the  co-efficients  of  ar  in 
succeeding  terms  may  be  found  ;  the  number  of  the  co-effi- 
cients being  one  less  than  the  number  of  roots  in  the  equation. 

Collecting  these  results,  we  have 

S,-\-AS,^mB=  (m  -  2)B, 
S,+AS,-\-BS\-{-mC=  (m  -  3)  C, 
S,+.qS,-{-BS2-{-  CSy+mD=  (m  -  4)D, 
&c.  &c. 

Transposing  and  uniting  terms, 

I.  S,-ir^=0, 
S^-^JlSy-\-2B=0, 
S,^JlS,-\-BS\-\-SC=0, 
S,Jr^^Ss-\-BS,Jf-CS,-{-4D=0, 

&c.  &c. 

Substituting  for  aS,,  ♦S'a,  S^,  &c.  their  values,  and  reducing, 

II.  S,=  -Jl, 
S,=     ^-2B, 

S,=  -JP-\-SAB-SC, 
S,=     ^'-4Jl'B-\-4AC+2B'-4D, 
&c.  &c. 

We  have  here  obtained  symmetrical  expressions  for  the 
sum  of  the  roots  of  an  equation,  the  sum  of  their  squares, 
the  sum  of  their  cubes,   &:c.  in  terms  of  the  co-efficients. 


EQUATIONS.  291 

By  transposing  the  terms  in  the  expressions  marked  I,  we 
have  the  following  values  of  Jl,  B,  C,  &c. 

III.  A=  -  S, 

&c.  &c. 

By  which  the  co-efficients  of  an  equation  may  be  found, 
from  the  sum  of  its  roots,  the  sum  of  their  squares,  the  sum 
of  their  cubes,  &c. 

Ex.  1.  Required  the  sum  of  the  roots,  the  sum  of  their 
squares,  and  the  sum  of  their  cubes,  in  the  equation 

x'-l0x'-^S5x'-50x-24=0, 

HeTeA=  - 10.  B=S5.  C=-50. 

Therefore  ^^=10 

^3  =  103+(3x  -  10x35)  -  (3x  -50)  =  100. 

2.  Required  the  terms  of  the  biquadi-atic  equation  in  which 
Si  =  l,  *Sa=39,  83=  -89,  and  the  product  of  all  the  roots 
after  their  signs  are  changed  is  -  30. 

Ans.  x'-a^-  19a;'^+49a; -  30=0.* 

*  See  Note  V. 


292  ALGEBRA. 


SECTION  XXI. 


APPLICATION  OF  ALGEBRA  TO  GEOMETRY. 


Art.  504.  It  is  often  expedient  to  make  use  of  the  alge- 
braic notation,  for  expressing  the  relations  of  geometrical 
quantities,  and  to  throw  the  several  steps  in  a  demonstration 
into  the  form  of  equations.  By  this,  the  nature  of  the  reason- 
ing is  not  aUered.  It  is  only  translated  into  a  different  lan- 
guage. Signs  are  substituted  for  words,  but  they  are  intend- 
ed to  convey  the  same  meaning.  A  great  part  of  the  de- 
monstrations in  Euclid,  really  consist  of  a  series  of  equa- 
tions, though  they  may  not  be  presented  to  us  under  the  al- 
^gebraic  forms.  Thus  the  proposition,  that  the  sum  of  the 
three  angles  of  a  triangle  is  equal  to  two  right  angles,  (Euc.  32. 
1.)  may  be  demonstrated,  either  in  common  language,  or  by 
means  of  the  signs  used  in  Algebra. 

Let  the  side  JlB,  of  the  triangle  ABC,  (Fig.  1.)  be  con- 
tinued to  D;  let  the  line  BE  be  parallel  io  AC;  and  let 
GHI  be  a  right  angle. 

The  demonstration,  in  words,  is  as  follows : 

1.  The  angle  EBD  is  equal  to  the  angle  BAC,  (Euc.  29.  1.) 

2.  The  angle  CBE  is  equal  to  the  angle  ACB. 

3.  Therefore,  the  angle  EBD  added  to  CBE,  that  is,  the 

angle  CBD,  is  equal  to  BAC  added  to  ACB. 

4.  If  to  these  equals,  we  add  the  angle  ABC,  the  angle  CBD 

added  to  ABC,  is  equal  to  BAC  added  to  ACB  and 
ABC. 


*  This  and   th«  following  section  are  to  be  read  after  the  Elements  of 
Geometry. 


APPLICATION  TO  GEOMETRY,  w  £93 

5.  But  CBD  added  to  .^BC,  is  equal  to  twice  GHT,  that  is, 

to  (wo  lig-lit  angles.  (Eiic.  13.  1.) 

6.  Therefore,  the  angles  BAC,  and  ACB,  and  ABC,  are  to- 

gether equal  to  twice  GHI,  or  two  right  angles. 

No\v  by  substituting  tlie  sign  -\-,  for  the  word  added,  or 
and,  and  the  character  —,  for  the  word  equal,  we  shall  have 
the  same  demonstration  in  the  followinc^  form. 

1.  By  Euclid  29.  1.  EBD=BAC 

2.  And  CBE=ACB 

3.  Add  the  two  equations  EBD-\-CBE=BAC-\-ACB 
4.-  Add  ABC  to  both  sides  CBD-\-ABC=:BAC-^ACB-{- 

ABC 

5.  But  by  Euclid  13.  1.  CBDJr^lBC=:2GHI 

6.  Make  the  4th  &  5th  equal  BAC^ACB-\-ABC=2GHL 

By  comparing,  one  by  one,  the  steps  of  these  two  demon- 
strations, it  will  be  seen,  that  they  are  precisely  the  same,  ex- 
cept that  they  are  differently  expressed.  The  algebraic  mode 
has  often  the  advantage,  not  only  in  being  more  concise  than 
the  other,  but  in  exhibiting  the  order  of  the  quantities  more 
distinctly  to  the  eye.  Thus,  in  the  fourth  and  fifth  steps  of 
the  preceding  example,  as  the  parts  to  be  compared  are 
placed  one  under  the  other,  it  is  seen,  at  once,  what  must  be 
the  new  equation  derived  from  these  two.  This  regular  ar- 
rangement is  very  important,  when  the  demonstration  of  a 
theorem,  or  the  resolution  of  a  problem,  is  unusually  compli- 
cated. In  ordinary  language,  the  numerous  relations  of  the 
quantities,  require  a  series  of  explanations  to  make  them  un- 
derstood ;  while  by  the  algebraic  notation,  the  whole  may  be 
placed  distinctly  before  us,  at  a  single  view.  The  disposi- 
tion of  the  men  on  a  chess-board,  or  the  situation  of  the  ob- 
jects in  a  landscape,  may  be  better  comprehended,  by  a 
glance  of  the  eye,  than  by  the  most  laboured  description  in 
words. 

505.  It  will  be  observed,  that  the  notation  in  the  example 
just  given,  differs,  in  one  respect,  from  that  which  is  general- 
ly used  in  algebra.  Each  quantity  is  represented,  not  by  a 
single  letter,  but  by  several.  In  common  algebra  w^hen  one 
letter  stands  immediately  before  another,  as  ab,  without  any 
character  betw^een  them,  they  are  to  be  considered  as  multi- 
plied  together. 


294  *  ALGEBRA. 

But  in  geometry,  AB  is  an  expression  for  a  single  line,  and 
not  for  the  product  of  A  into  B.  Multiplication  is  denoted, 
either  by  a  point  or  by  the  character  x.  The  product  of 
AB  into  CD,  is  ABCD,  or  ABxCD. 

506.  There  is  no  impropriety,  however,  in  representing  a 
geometrical  quantity  by  a  single  letter.  We  may  make  h 
stand  for  a  line  or  an  angle,  as  well  as  for  a  number. 

If,  in  the  example  above,  we  put  the  angle 

EBD=a,  ACB=d,  JlBC=h, 

BJlC=b,  CBD=g,  GHI=l; 

CBE=c, 

the  demonstration  will  stand  thus  ; 

1.  By  Euclid,  29.  1.  a=6 

2.  And  c=d 

3.  Adding  the  two  equations,  a-\-c=g=b-\-d 

4.  Adding  h  to  both  sides,  g-\-h=b-\'d-\-h 

5.  By  Euclid  13.  1.  g+h=2l 

6.  Making  the  4th  and  5th  equal,  b-\-d-{-h=2l. 

This  notation  is,  apparently,  more  simple  than  the  other  ; 
but  it  deprives  us  of  what  is  of  great  importance  in  geometri- 
cal demonstrations,  a  continual  and  easy  reference  to  the 
figure.  To  distinguish  the  two  methods,  capitals  are  gener- 
ally used,  for  that  which  is  peculiar  to  geometry  ;  and  small 
letters,  for  that  which  is  properly  algebraic.  The  latter  has 
the  advantage  in  long  and  complicated  processes,  but  the 
other  is  often  to  be  preferred,  on  account  of  the  facility  with 
which  the  figures  are  consulted. 

507.  If  a  line,  whose  length  is  measured  from  a  given 
point  or  line,  be  considered  positive ;  a  line  proceeding  in  the 
opposite  direction  is  to  be  considered  negative.  If  AB  (Fig. 
2.)  reckoned  from  DE  on  the  right,  is  positive  ;  AC  on  the 
left  is  negative. 

A  line  may  be  conceived  to  be  produced  by  the  motion  of 
a  point.  Suppose  a  point  to  move  in  the  direction  of  AB, 
and  to  describe  a  line  varying  in  length  with  the  distance  of 
the  point  from  A.  While  the  point  is  moving  towards  B,  its 
distance  from  A  will  increase.  But  if  it  move  from  B  to- 
wards C,  its  distance  from  A  will  diminish,  till  it  is  reduced 


APPLICATION  TO  GEOMETRY.  295 

to  rK)(hing,  and  then  will  increase  on  the  opposite  side.  As 
that  which  increases  the  distance  on  the  right,  diminishes  it 
on  the  left,  the  one  is  considered  positive,  and  the  other  nega- 
tive.    See  Arts.  59,  60. 

Hence,  if  in  the  course  of  a  calculation,  the  algebraic 
value  of  a  line  is  found  to  be  negative;  it  must  be  measured 
in  a  direction  opposite  to  that  wliich,  in  the  same  process, 
has  been  considered  positive.    (Art.  197.) 

508.  In  algebraic  calculations,  there  is  frequent  occasion 
for  multiplication  J  division,  involution,  &c.  But  how,  it  may 
be  asked,  can  geometrical  cpiantities  be  multiplied  into  each 
other?  One  of  the  factors,  in  multiplication,  is  always  to  be 
considered  as  3i  number.  (Art.  91.)  The  operation  consists  in 
repeating  the  multiplicand  as  many  times  as  there  are  units 
in  the  multiplier.  How  then  can  a  line,  a  surface,  or  a  solidy 
become  a  multipher  ? 

To  explain  this  it  will  be  necessary  to  observe,  that  when- 
ever one  geometrical  quantity  is  multiplied  into  another, 
some  particular  extent  is  to  be  considered  the  unit.  It  is  imma- 
terial what  this  extent  is,  provided  it  remains  the  same,  in 
different  parts  of  the  same  calculation.  It  may  be  an  inch, 
a  foot,  a  rod,  or  a  mile.  If  an  inch  is  taken  for  the  unit, 
each  of  the  lines  to  be  multiplied,  is  to  be  considered  as  made 
up  of  so  many  parts,  as  it  contains  inches.  The  multiplicand 
will  then  be  repeated,  as  many  times,  as  there  are  units  in 
the  multiplier.  If,  for  instance,  one  of  the  lines  be  a  foot 
long,  and  the  other  half  a  foot ;  the  factors  will  be,  one  12 
inches,  and  the  other  6,  and  the  product  will  be  72  inches. 
Though  it  would  be  absurd  to  say  that  one  line  is  to  be  re- 
peated as  often  as  another  is  long  ;  yet  there  is  no  impropriety 
in  saying,  that  one  is  to  be  repeated  as  many  times,  as  there 
ore  feet  or  rods  in  the  other.  This,  the  nature  of  a  calcula- 
tion often  requires. 

509.  If  the  line  which  is  to  be  the  multiplier,  is  only  a 
part  of  the  length  taken  for  the  unit ;  the  product  is  a  like 
part  of  the  multiplicand.  (Art.  90.)  Thus,  if  one  of  the 
factors  is  6  inches,  and  the  other  half  an  inch,  the  product  is 
3  inches. 

510.  Instead  of  referring  to  the  measures  in  common  use, 
as  inches,  feet,  &c.  it  is  often  convenient  to  fix  upon  one  of 
the  lines  in  a  figure,  as  the  unit  with  which  to  compare  all  the 
others.     When  there  are  a  number  of  Hues  drawn  within 


296  ALGEBRA. 

and  about  a  circle,  tlie  radius  is  commonly  taken  for  the  unit. 
This  is  particularly  the  case  in  trigonometrical  calculations. 

511.  The  observations  which  have  been  made  concerning 
lines,  may  be  applied  to  surfaces  and  solids.  There  may  be 
occasion  to  multiply  the  area  of  a  figure,  by  the  number  of 
inches  in  some  given  line. 

But  here  another  difficulty  presents  itself.  The  product 
of  two  lines  is  often  spoken  of,  as  being  equal  to  a  surface  ; 
and  the  product  of  a  line  and  a  surface,  as  equal  to  a  solid. 
Thus  the  area  of  a  parallelogram  is  said  to  be  equal  to  the 
product  of  its  base  and  height ;  and  tlie  solid  contents  of  a 
cylinder,  are  said  to  be  equal  to  the  product  of  its  length  into 
the  area  of  one  of  its  ends.  But  if  a  line'  has  no  breadth, 
how  can  the  multiplication,  that  is  the  repetition,  of  a  line 
produce  a  surface  1  And  if  a  surface  has  no  thickness,  how 
can  a  repetition  of  it  produce  a  solid  ] 

If  a  parallelogram,  represented  on  a  reduced  scale  by 
ABCD,  (Fig.  3.)  be  five  inches  long,  and  three  inches  wide  ; 
the  area  or  surface  is  said  to  be  equal  to  the  product  of  5  into 
3,  that  is,  to  the  number  of  inches  in  AB,  multiplied  by  the 
number  in  BC.  But  the  inches  in  the  lines  AB  and  ^C  are 
linear  inches,  that  is,  inches  in  length  only ;  w^hile  those 
which  compose  the  surface  JlC  are  superficial  or  square 
inches,  a  different  species  of  magnitude.  How^  can  one  of 
these  be  converted  into  the  other  by  muUiplication,  a  process 
which  consists  in  repeating  quantities,  without  changing 
their  nature  ? 

512.  In  ansAvering  these  in.quiries,  it  must  be  admitted, 
that  measures  of  length  do  not  belong  to  the  same  class  of 
magnitudes  with  superficial  or  solid  measures  ;  and  that  none 
of  the  steps  of  a  calculation  can,  })roperly  speaking,  trans- 
form the  one  into  the  other.  But,  though  a  line  cannot  be- 
come a  surface  or  a  solid,  yet  the  several  measuring  units  in 
common  use  are  so  adapted  to  each  other,  that  squares, 
cubes,  &c.  are  bounded  by  lines  of  the  same  name.  Thus 
the  side  of  a  square  inch,  is  a  linear  inch  ;  that  of  a  square 
rod,  a  linear  rod,  &c.  The  length  of  a  linear  inch  is,  there- 
fore, the  same  as  the  length  or  breadth  of  a  square  inch. 

If  then  several  square  inches  are  placed  together,  as  from 
Q  to  R,  (Fig.  3.)  the  number  of  them  in  the  parallelogram 
OR  is  the  same  as  the  number  of  linear  inches  in  the  side 
QR :  and  if  we  know  the  length  of  this,  we  have  of  course 


APPLICATION  TO  GEOMETRY.  297 

the  area  of  the  parallelogram,  which  is  here  supposed  to  be 
one  inch  wide. 

But,  if  the  breadth  is  several  inches,  the  larger  parallelo- 
gram contains  as  many  smaller  ones,  each  an  inch  wide,  as 
there  are  inclies  in  the  whole  breadtli.  Thus,  if  tlie  paral- 
lelogram c^C  (Fig.  3.)  is  5  inches  long,  and  3  inches  broad, 
it  may  be  divided  into  three  such  parallelograms  as  OR.  To 
obtain,  then,  the  number  of  squares  in  the  large  parallelo- 
gi-ain,  we  have  only  to  multiply  the  number  of  squares  in 
one  of  the  small  paraUelograms,  into  the  number  of  such 
parallelograms  contained  in  the  whole  figure.  But  the  num- 
ber of  square  inclies  in  one  of  the  small  parallelograms  is 
equal  to  the  mnnbcr  of  linear  inches  in  the  length  JIB.  And 
the  number  of  small  parallelograms,  is  equal  to  the  number 
of  linear  inches  iji  the  breadth  BC.  It  is  therefore  said  con- 
cisely, that  the  area  of  the  parallelogram  is  equal  to  the  length 
multiplied  into  the  breadth. 

513.  We  hence  obtain  a  convenient  algebraic  expression, 
for  the  area  of  a  right-angled  parallelogram.  If  two  of  the 
sides  perpendicular  to  each  oiher  are  AB  and  BC,  the  expres- 
sion for  the  area  is  ABx^C  ;  that  is,  putting  a  for  the  area, 

a=ABxBC. 

It  must  be  understood,  how^ever,  that  when  AB  stands  for 
a  line,  it  contains  only  linear  measuring  units ;  but  Avhen  it 
enters  into  the  expression  for  the  area,  it  is  supposed  to  con- 
tain superficial  units  of  the  same  name.  Yet  as,  in  a  given 
length,  the  number  of  one  is  equal  to  that  of  the  other,  they 
may  be  represented  by  the  same  letters,  without  leading  to 
error  in  calculation. 

514.  The  expression  for  the  area  may  be  derived,  by  a 
method  more  simple,  but  less  satisfactory  perhaps  to  some, 
from  the  principles  which  have  been  stated  concerning  vari- 
able quantities,  in  the  13th  section.  Let  a  (Fig:  4.)  represent 
a  square  inch,  foot,  rod,  or  other  measuring  unit ;  and  let  h 
and  I  be  tw^o  of  its  sides.  Also,  let  A  be  the  area  of  any 
right-angled  parallelogram,  B  its  breadth,  and  L  its  length. 
Then  it  is  evident,  that,  if  the  breadth  of  each  were  the 
same,  the  areas  w^ould  be  as  the  lengths  ;  and,  if  the  length 
of  each  were  the  same,  the  areas  would  be  as  the  breadths. 

That  is,         A\a\'.L:l,  when  the  breadth  is  given  ; 
And  A  :  aw  B  :  b,  when  the  length  is  given ; 

38 


298  ALGEBRA. 

Therefore,  (Art.  420.)  A  .  a.  :  B^L  \hl,  when  both  vary. 
That  is,  the  area  is  as  the  product  of  tiie  length  and  breadth. 

515.  Hence,  in  quoting  the  Elements  of  Eudid,  the  term 
product  is  frequently  substituted  for  rectangle.  And  what- 
ever is  there  proved  concerning  the  equality  of  certain  rect- 
angles, may  be  applied  to  the  product  of  the  lines  which 
contain  the  rectangles.* 

516.  The-  area  of  an  oblique  parallelogram  is  also  obtained, 
by  multiplying  the  base  into  the  perpendicular  height.  Thus 
the  expression  for  the  area  of  the  parallelogram  ABJ^MiY'ig. 
5.)  is  MJ^x^D  or  JlBxBC.  For  by  Art.  513,  ABxBC 
is  the  area  of  the  right-angled  parallelogram  JIB  CD  ;  and 
by  Euclid  36,  l,t  parallelograms  upon  equal  bases,  and  be- 
tween the  same  parallels,  are  equal ;  that  is,  JIB  CD  is  equal 
io  JIB  KJ\L 

517.  The  area  of  a  square  is  obtained,  by  multiplying  one 
of  the  sides  into  itself.     Thus  the  expression  for  the  area  of 

2 

the  square  JIC,  (Fig.  6,)  is  AB,  that  is, 

a=z~f]f. 
For  the  area  is  coual  to  ABxDC.     (Art.  513.) 
But  AB=BC,  therefore,  ABxBC=ABx^^B=AB. 

518.  The  area  of  a  inangle  is  equal  to  half  the  product  of 
the  base  and  height.  .Thus  the  area  of  the  triangle  ABG, 
(Fig.  7.)  is  equal  to  half  AB  into  GHoy  its  equal  BC,  that  is, 

a=iABxBC. 
For  the  area  of  the  parallelogram  ABCD  is  ABxBC, 
(Art.  513.)  And  by  Euc.  41,  l,:j:  if  a  parallelogram  and  a  tri- 
angle are  upon  the  same  base,  and  between  the  same  paral- 
lels, the  triangle  is  half  the  parallelogram. 

159.  Hence,  an  algebraic  expression  may  be  obtained  for  the 
area  of  any  figure  whatever,  whicii  is  bounded  by  right  lines. 
For  every  such  figure  may  be  divided  into  triangles. 


*  See  Note  W. 

ILegendre's  Geometry,  American  Edition,  Art.  166. 
L€gcndre,  163. 


APPLICATION  TO  GEOMETRY.  299 

Thus  the  right-Hned  figure 
ABODE  (Fig.  8,)  is  composed  of  the  triangles 
ABC,  ACE,  and  ECD. 

The  area  of  the  triangle  ABC=\ACxBL, 

That  of  the  triangle  ACE=l,ACxEH, 

That  of  the  triangle  ECD=\ECxDG. 

The  area  of  the  whole  figure  is,  therefore,  equal  to 
{\ACxBL)-\-{h^CxEH)-^r{iECxBG), 

The  explanations  in  the  preceding  articles  contain  the 
first  principles  of  the  mensuration  of  superficies.  The  object  of 
introducing  the  subject  in  this  place,  however,  is  not  to  make 
a  practical  application  of  it,  at  present  ;  but  merely  to  show 
the  grounds  of  the  method  of  representing  geometrical  quan- 
tities in  algebraic  language. 

520.  The  expression  for  the  superficies  has  here-  been  de- 
rived from  that  of  a  line  or  lines.  It  is  frequently  necessary 
to  reverse  this  order  ;  to  fmd  a  side  of  a  figure,  from  knowing 
its  area.  - 

If  the  number  of  square  inches  in  the  parallelogram 
ABCD  (Fig.  3.)  whose  breadth  J5C  is  3  ii>ches.,  be  divided 
by  3  ;  the  quotient  will  be  a  parallelogram  ABtlF,  one  inch 
wide,  and  of  the  same  length  with  the  larger  one.  But  the 
length  of  the  small  parallelogram,  is  the  length  of  its  side 
AB.  The  number  of  square  inches  in  one  is  the  same,  as 
the  number  of  linear  inches  in  the  other.  (Art.  512.)  If 
therefore,  the  area  of  the  large  parallelogram  be  represented 

by  a,  the  side  AB= ,  that  is,  the  length  of  a  parallelogram 

BC 
is  found  by  dividing  the  area  by  the  breadth. 

52 L   If  a  be  put  for  the  area  of  a  square  whose  side  is  AB, 

Then  by  Art.  517  a=AB'' 

And  extracting  both  sides  ^a=AB. 

That  is,  the  side  of  the  square  is  f&und,  by  extracting  the 
square  root  of  the  number  of  measuring  units  in  its  area, 

522.  If  AB  be  the  base  of  a  triangle  and  BC  its  perpen- 
dicular height ; 


300  ALGEBRA, 

Then  by  Art.  518,  a=\BCx^B 

And  dividing  by  \BC,  -^=AB. 

That  is,  the  base  of  a  triangle  is  found,  by  dividing  the  area 
by  half  the  height. 

523.  As  a  surface  is  expressed,  by  the  product  of  its  length 
and  breadtli ;  the  contents  of  a  solid  may  be  expressed,  hy 
the  product  of  its  length,  breadth  and  deptli.  It  is  necessary 
to  bear  in  mind,  that  the  measuring  unit  of  solids,  is  a  cube  ; 
and  that  the  side  of  a  cubic  incii,  is  a  square  inch  ;  the  side 
of  a  cubic  foot,  a  square  foot,  &c. 

Let  ABCD  (Fig.  3.)  represent  tlie  base  of  a  parallelopi- 

fed,  5  inches  long,  three  inches  broad,  and  one  inch  deep, 
t  is  evident  there  must  be  as  many  cubic  inches  in  the  solid, 
as  there  are  square  inches  in  its  base.  And,  as  the  product  of 
the  lines  AB  and  .SC  gives  the  area  of  this  base,  it  gives,  of 
course,  the  contents  of  the  solid.  But  suppose  that  the  depth 
of  the  parallelopiped,  instead  of  being  one  inch,  is  four  inches. 
Its  contents  must  be  four  times  as  great.  If,  then,  the 
length  be  ^QB,  t]ie  breadth  BC,  and  the  depth  CO,  the  ex- 
pression for  the  solid  contents  will  be, 
JlBxBCxCO. 

524.  By  means  of  the  algebraic  notation,  a  geometrical 
demonstration  may  often  be  rendered  much  more  simple  and 
concise,  than  in  ordinary  language.  The  proposition,  (Euc. 
4.  2.)  that  when  a  straight  line  is  divided  into  two  parts,  the 
square  of  the  whole  line  is  equal  to  the  squares  of  the  two 
parts,  together  with  tv/ice  the  product  of  the  parts,  is  demon- 
strated, by  inv^olving  a  binomial. 

Let  the  side  of  a  square  be  represented  by  s ; 
And  let  it  be  divided  into  two  parts,  a  and  b. 
By  the  supposition,  s=a-\-b 

And  squaring  both  sides,  s''^=a^-\-2ab-{-b'^. 

That  is,  s"^  the  square  of  the  whole  line,  is  equal  to  o^  and 
b\  the  squares  of  the  two  parts,  together  with  2ab,  twice  the 
product  of  the  parts. 

525.  The  algebraic  notation  may  also  be  applied,  with 
great  advantage,  to  the  solution  of  geometrical  problems.  In 
doing  this,  it  will  be  necessary,  in  the  first  place,  to  raise  an 


GEOMETRICAL  PROBLEMS.  301 

algebraic  equation,  from  the  geometrical  relations  of  the 
quantities  given  and  required ;  and  then  by  the  usual  reduc- 
tions, to  find  the  value  of  the  unknown  quantity  in  this  equa- 
tion.    See  Art.  192. 

Prob.  1.  Given  the  base,  and  the  sum  of  the  hypothenuse 
and  perpendicular,  of  the  right  angled  triangle,  ABC,  (Fig. 
9.)  to  find  the  perpendicular. 

Let  the  base  JlB=h 

The  perpendicular  BC=x 

The  sum  of  hyp.  and  perp.       x-\-*.flC=a 
Then  transposing  a:,      •  AC=a-  x 

1.  By  Euclid  47.  1,*         BC-\-AB=AC 

2.  That  is,  by  the  notation,  x''-]-b^={a- xy=a''-2ax-{-x\ 

Here  we  have  a  common  algebraic  equation,  containing 
only  one  unknown  quantit}'.  The  reduction  of  this  equa- 
tion in  the  usual  manner,  will  give 

x= —  BC\  the  side  required. 

2a 

The  solution,  in  letters,  will  be  the  same  for  any  right 
angled  triangle  whatever,  and  may  be  expressed  in  a  gene- 
ral theorem,  thus  ;  '  In  a  right  angled  triangle,  the  perpendi- 
cular is  equal  to  the  square  of  the  sum  of  the  hypothenuse 
and  perpendicular,  diminished  by  the  square  of  the  base,  and 
divided  by  twice  the  sum  of  the  hypothenuse  and  perpendi- 
cular.' 

It  is  applied  to  particular  cases  by  substituting  numbers,  for 
the  letters  a  and  b.  Tlius  if  the  base  is  8  feet,  and  the  sum 
of  the   hypothenuse  and   perpendicular  16,  the  expression 

^LJI — becomes  — Z — =6,  the  perpendicular  ;  and  this  sub- 

2a  2x16 

traded  from  16,  the  sum  of  the  hypothenuse  and  perpendi- 
cular, leaves  10,  the  length  of  the  hypothenuse. 

Prob.  2.  Given  the  base  and  the  difference  of  the  hypothe- 
nuse and  perpendicular,  of  a  right  angled  triangle,  to  find  the 
perpendicular. 

*  Le^endre,  186. 


302  ALGEBRA. 

Let  tlie  base  AB  (Fig.  10.)=fc=20 

The  perpendicular,  BC—x 

The  given  difference,  =d=10. 

Then  will  the  hypothenuse  AC—x-\-d. 

Then 


1.  By  Euclid  47.  1,  JiC  =AB^BC 

2.  That  is,  by  the  notation,  {x^dY^h^-^-r} 

3.  Expanding  {x-^d)\  x'-^Ux^d^=h''-^x'' 

4.  Therefore  ,  x=lzl.  =  \b. 

2d 

Prob.  3.  If  the  hypothenuse  of  a  right  angled  triangle  is 
30  feet,  and  the  difference  of  the  other  two  sides  6  feet,  v/hat 
is  the  length  of  the  base  1  Ans.  24  fee* 

Prob.  4.  If  the  hypothenuse  of  a  right  angled  triangle  is 
50  rods,  and  the  base  is  to  the  perpendicular  as  4  to  3,  what 
is  the  length  of  the  perpendicular  ?  '  Ans.  30. 

Prob  5.  Having  the  perimeter  and  the  diagonal  of  a  par- 
allelogram ABCD,  (Fig.  11.)  to  find  the  sides. 

Let  the  diagonal  AC=h=lO 

The  side  AB^x 

Half  the  perimeter  BC-]-AB=BC-\-x=b  =14 
Then  by  transposing  x,  BC=b-x 

By  Euclid  47.  1,  AB-\-'BC  ='Jc 

That  is,  x''-\-{b-xY=h^ 

Therefore  x=i  b±VW+W^^W=  8- 

Here  the  side  AB  is  found ;  and  the  side  j5C  is  equal  to 

h-X=U-S=:6. 

Prob.  6.  The  area  of  a  right  angled  triangle  ABC  (Fig. 
12,)  being  given,  and  the  sides  of  a  parallelogram  inscribed 
in  it,  to  find  the  side  BC. 

Let  the  given  area  =ia,         DE=BF=b         ^ 

EB=DF=d,  BC=x.        [ 

Then  by  the  figure,      CF=BC  -  BF=x  -b.) 


GEOMETRICAL   PROBLEMS.  303 

1.  By  similar  triangles,  CF:DF::BC:  .W 

2.  That  is,  •  x-b:  d::x  :  JIB 

3.  Therefore,  dx={x^h)x^W 

4.  By  Art.  518,  a=ABx\BC=ABx\x 

5.  Dividing  by  Jar,  —=AB 

X 

6.  Therefore  dx={x-h)  X—=-^ci-^ 

X  X 


7.  And  :.=«+^«;-!^*=BC. 

Prob.  7.  The  three  sides  of  a  right  angled  triangle,  ..5 J5C, 
(^Fig.  13.)  being  given,  to  find  the  segments  made  by  a  per- 
pendicular, dra\\^n  from  the  right  angle  to  the  hypothenuse. 

The  perpendicular  will  divide  the  original  triangle,  into 
two  right  angled  triangles,  BCD  and  ABD.  (Euc.  8  6.)* 

1.  By  Euc.  47.  1,  BD +CdL5C 

2.  By  the  figure,  CD=.^C-AD 

3.  Squar.  both  sides,  C"D=  {AC- AX)y 

4.  Therefore,  BD+(.^C  -  AD)=^ 

5.  Expanding,  ^-\-M-2AC.KD-\-KD=BC 

6.  Transposing,  ^=BC-AC-^%AC,KD-JS^ 

7.  By  Euc.  47.  1.  BdLZB- AD 

8.  Mak.  6th  &  7th  eq.  BC- AC  J^2  A  CAT)  =^AB 

A    mi,      r  An     ^B^AC-BC 

9.  Therefore  AD= -^^-77^^ 

The  unknown  lines,  to  distinguish  them  from  those  which 
are  known,  are  here  expressed  by  Roman  letters. 

Prob.  8.  Having  the  area  of  a  parallelogram  DEFG  (Fig., 
14,)  inscribed  in  a  given  triangle,  ABC,  to  find  the  sides  of 
the  parallelogram. 


Legendre,  213. 


304  ALGEBRA. 

Draw  CI  perpendiculat  to  AB.  By  supposition,  DO  is 
parallel  to  JIB.     Therefore, 

The  triangle  CHG,  is  similar  to  CIB  > 
And  CDG,  to  CAB  5 

Let  CI=d  DG=x  > 

AB=b  The  given  area  =a) 

L  By  similar  triangles,  CB  :  CO  :  :AB  .  DG 

2.  And  CB  :  CG  : :  CI :  CH 

3.  By  equal  ratios,  (Art.  384.)  AB  :  DG.:  CI .  CH 

4.  Therefore  P^^L=CH 

JiB 

5.  By  the  figure,  CI-  CH=IH=DE 

6.  Substituting  for  CH,  CI-^^^J^^=DE 

JIB 

7.  That  is,  d-^=DE 

0 

8.  By  Art.  513,  a=DGxDE=xX{d-^\ 

9.  That  is,  a=dx-?f. 

b 

10.  This  reduced  ffives  x=-^  ^  /  —-^—HQ 

2-V  (4    d 

The  side  BE  is  found,  by  dividing  the  area  b}''  J)G, 

Prob.  9.  ThroiT^h  a  given  point,  in  a  given  circle,  so  to 
draw  a  right  line,  that  its  parts,  between  the  point  and  the 
periphery,  shall  have  a  giveii  difference. 

In  the  circle  AQBR,  (Fig.  15.)  let  P  be  a  given  point,  in 
the  diameter  JIB. 

LeiAP=a,  PR=x, 

BPz=bi  The  given  difference =(^, 

Then  will  PQ=a;-l-rf. 


GEOMETRICAL   PROBLEMS.  305 

L  By  Euc.  35.  3.*  PRxPQ=^PxBP 

2.  That  is,  .TX{x-\-d)=axb 

3.  Or,  x'-\-dx=:ab 

4.  Completing  the  square,  x'''-\-dx-\-ld^=\d^-{-ab. 

5 .  Extract,  and  transp.  3:=-^ d±\/ld''-\-ab  =  PR. 

With  a  httle  practice,  the  learner  may  very  much  abridge 
these  solutions,  and  others  of  a  similar  nature,  by  reducing 
several  steps  to  one. 

Prob  10.  If  the  sum  of  two  of  the  sides  of  a  triangle  be 
1 155,  the  length  of  a  perpendicular  drawn  from  the  angle  in- 
cluded between  these  to  the  third  side  be  300,  and  the  differ- 
ence of  the  segments  made  by  the  perpendicular,  be  495  ; 
what  are  the  lengths  of  the  three  sides  1 

Ans.  945,  375,  and  780. 

Prob.  11.  If  the  perimeter  of  a  riglit  angled  triangle  be 
720,  and  tbe  perpendicular  falling  from  the  right  angle  on 
the  hypotlieiuise  be  144  ;  what  are  the  lengths  of  the  sides  1 

Ans.  300,  240,  and  180. 

Prob.  12.  The  difference  between  the  diagonal  of  a  square 
and  one  of  its  sides  being  given,  to  find  the  length  of  the 
sides. 

If  x=  the  side  rerpiired,  and  d=  the  given  difference ; 
Then  x=d-\-d\/2. 

Prob.  14.  Tbe  base  and  perpendicular  height  of  an)^  plane 
triangle  being  given,  to  find  the  side  of  a  square  inscribed  in 
the  triangle,  and  standing  on  the  base,  in  the  same  manner 
as  the  parallelogram  Z)£FG,  on  the  base  *^B,  (Fig.  14.) 

If  a::;=  a  side  of  the  square,  b=  the  base,  and  h=z  the 
height  of  the  triangle  ; 

Thena;^J^. 

Prob.  15.  Two  sides  of  a  triangle,  and  a  line  bisecting  the 
included  angle  being  given  ;  to  find  tbe  length  of  the  base 
or  third  side,  upon  which  the  bisecting  line  falls. 


*  Legendre  224. 
39 


306  ,  Al^iJiLBKA. 

If  a;=  the  base,  a=  one  of  the  given  sides,  c=  the  other, 
and  b=  the  bisecting  line  ; 

Then  x=  (a-Jrc)  X  ^"^'"^^ 

Prob.  16.  If  the  hypothenuse  of  a  right  angled  triangle 
be  35,  and  the  side  of  a  square  inscribed  in  it,  in  the  same 
manner  as  the  parallelogram  i?£i)i^,  (Fig.  12.)  be  12  ;  what 
are  the  lengths  of  the  other  two  sides  of  the  triangle  1 

Ans.  28,  and  21. 

Prob.  17.  The  nmiiber  of  feet  in  the  perimeter  of  a  right 
angled  triangle,  is  equal  to  the  number  of  square  feet  in  the 
area  ;  and  the  base  is  to  the  perpendicular  as  4  to  3.  Re- 
quired the  length  of  each  of  the  sides. 

Ans.  6,  8,  and  10. 

Prob.  18.  A  grass  plat  12  rods  by  18,  is  surrounded  by  a 
gravel  walk  of  uniform  breadth,  whose  area  is  equal  to  that 
of  the  grass  plat.     What  is  the  breadth  of  the  gravel  w^alk  ? 

Prob.  19.  The  sides  of  a  rectangular  field  are  in  the  ratio 
of  6  to  5  ;  and  one  sixth  of  the  area  is  125  square  rods. 
What  are  the  lengths  of  the  sides  ? 

Prob.  20.  There  is  a  right  angled  triangle,  the  area  of 
which  is  to  the  area  of  a  given  parallelogram  as  5  to  8.  The 
shorter  side  of  each  is  60  rods,  and  the  other  side  of  the  tri- 
angle adjacent  to  the  right  angle,  is  equal  to  the  diagonal  of 
the  parallelogram.     Required  the  area  of  each  ? 

Ans.  4800  and  3000  square  rods. 

Prob.  21.  There  are  two  rectangular  vats,  the  greater  of 
which  contains  20  cubic  feet  more  than  the  other.  Their 
capacities  are  in  the  ratio  of  4  to  5  ;  and  their  bases  are 
squares,  a  side  of  each  of  which  is  equal  to  the  depth  of  the 
other  vat.     Required  the  depth  of  each  1 

Ans.  4  and  5  feet. 

Prob.  22.  Given  the  lengths  of  three  perpendiculars, 
drawn  from  a  certain  point  in  an  equilateral  triangle,  to  the 
three  sides,  to  find  the  length  of  the  sides. 

If  a,  b,  and  c,  be  the  three  perpendiculars,  and  x=  half 
the  length  of  one  of  the  sides  ; 

Then  ,T=i!+M:f. 
V3 


GEOMETRICAL  PROBLEMS.  307 

Prob.  23.  A  square  public  green  is  surrounded  by  a  street 
of  uniform  breadth.  The  side  of  the  square  is  3  rods  less 
than  9  times  the  breadth  of  the  street ;  and  the  number  of 
square  rods  in  the  street,  exceeds  the  number  of  rods  in  the 
perimeter  of  the  square  by  228.  What  is  the  area  of  the 
square  ]  Ans.  576  rods. 

Prob.  24.  Given  the  lengths  of  two  lines  drawn  from  the 
acute  angles  of  a  right  angled  triangle,  to  the  middle  of  the 
opposite  sides  :  to  find  the  lengths  of  the  sides. 

If  x=  half  the  base,  y=z  half  the  perpendicular,  and  a 
and  h  equal  the  two  given  lines  ; 


*  See  Note  X. 


308  ALGEBRA. 


SECTION  XXil 


EQUATIONS  OF  CURVES. 


Art.  526.  IN  the  preceding  section,  algebra  has  been 
appUed  to  geometrical  figures,  bounded  by  right  lines.  Its  aid 
is  required  also,  in  investigating  the  nature  and  relations  of 
curves.  The  advances  which  in  modern  times  have  been 
made  in  this  department  of  geometry,  are,  in  a  great  measure, 
owing  to  the  metiiod  of  expressing  the  distinguishing  proper- 
ties of  the  different  kinds  of  lines,  in  the  form  of  equations. 
To  understand  the  principles  on  which  inquiries  of  this  sort 
are  conducted,  it  is  necessary  to  become  famihar  with  the 
plan  of  hotation  which  has  been  generally  agreed  upon. 

527.  The  positions  of  the  several  points  in  a  curve  drawn  on 
a  plane,  are  determined,  by  taking  the  distance  of  each  from  tioo 
right  lines  perpendicular  to  each  other. 

Let  the  lines  AF  and  AG  (Fig.  16.)  be  perpendicular  to 
each  other.  Also,  let  the  hues  DB,  D'B',  D"B"  be  perpen- 
dicular to  AF;  and  tlie  lines  CD,  CD',  C"D",  perpendicu- 
lar to  AG.  Then  the  position  of  the  point  Z>  is  known,  by 
the  length  of  the  lines  BD  and  CD.  In  the  same  manner, 
the  point  D'  is  known  by  t  he  lines  B'D'  and  CD' ;  and  the 
point  D",  by  the  lines  B'D'  and  CD".  The  two  lines 
which  are  thus  drawn,  from  any  point  in  the  curve,  are,  to- 
gether, called  ths  co-orrfma^es  belonging  to  that  point. 

But,  as  there  is  frequent  occasion  to  speak  of  each  of  the 
lines  separately,  one  of  them  for  distinction's  sake,  is  called 
an  ordinate,  and  the  otlier,  an  abscissa.  Tlius  BD  is  the  or- 
dinate of  the  point  D,  and  CD,  or  its  equal  AB,  the  abscissa 
of  the  same  point.  It  is,  generally,  most  convenient  to  take 
the  abscissas  on  the  line  AF,  as  AB  is  equal  to  CD,  AB' 
to  CD',  and  AB"  to  CD''.      Euc.  33.  1.     The  lines  Ah 


EQUATIONS   OF  CURVES.  309 

andc^G,  to  which  the  co-ordinates  are  drawn,  are  called  tlie 
axes  of  the  co-ordinates. 

528.  If  co-ordinates  could  be  drawn  to  every  point  in  a 
curve,  and,  if  the  relations  of  the  several  abscissas  to  their 
corresponding  ordinates  could  be  expressed  by  an  equation  ; 
the  position  of  each  point,  and  consequently,  the  nature  of 
the  curve,  would  be  determined.  Many  important  proper- 
ties of  the  figure  might  also  be  discovered,  merely  by  tin-ow- 
ing the  equation  into  diOerent  forms,  by  transposing,  dividing, 
involving,  &:c.  But  the  number  of  points  in  a  line  is  unlim- 
ited. It  is  impossible,  therefore,  actually  to  draw  co-ordi- 
nates to  every  one  of  them.  Still  there  is  a  way  in  which  an 
equation  may  be  obtained,  that  shall  be  applicable  to  all  the 
parts  of  a  curve.  This  is  effected  by  making  the  equation 
depend  on  some  propert}",  which  is  common  to  every  pair  of  co- 
ordinates. In  explaining  this,  it  will  be  proper  to  begin  Vvith 
a  straight  line,  instead  of  a  curve. 

Let  AH  (Fig.  17.)  be  a  line  from  which  co-ordinates  are 
drawn,  on  the  axes  AF  and  AG  perpendicular  to  each  other. 
And  let  the  angle  FAH  be  such,  that  the  abscissa  CD  or  AB 
shall  be  equal  to  twice  the  ordinate  BD. 

The  triangles  ABD,  AB'U,  AB'U'  &c.  are  all  similar. 
(Euc.  29.  1.)*  Therefore, 

AB  :  BD::  AB'  :  BD' :  :  AB'^  :  B"U\ 
And  if  AB=  2BD,  iheiiAB'=2B'D\  and  A B'' = 2 B^'D'',  Sic. 

That  is,  each  abscissa  is  equal  to  twice  the  corresponding 
ordinate.  But,  instead  of  a  separate  equation  for  each  pair 
of  co-ordinates,  one  will  be  sufficient  for  the  whole.  Let  x 
represent  any  one  of  the  abscissas,  and  y,  the  ordinate  be- 
longing to  the  same  point.     Then, 

x=2y,  oi-y=ix. 
This  is  an  equation  expressing  the  ratio  of  the  co-ordinates 
of  the  line  AH  to  each  other.  It  differs  from  a  common 
equation  in  this,  that  x  and  y  have  no  determinate  magni- 
tude. The  only  condition  which  limits  them  is,  tlmt  they 
shall  be  the  abscissa  and  ordinate  of  the  same  point. 

If  x=AB,  then  y=BD 

If  x=zAB\  y  =  B'D' 

If  x=^AB'\  y=B"U',  &LC. 


Legendre,  66. 


310  ALGEBRA. 

From  this  it  is  evident,  that,  if  one  of  the  co-ordinates  be 
taken  of  any  particular  length,  the  other  will  be  given  by  the 
equation.  If,  for  instance,  the  abscissa  xhe  two  inches  long, 
the  ordinate  y,  which  is  half  x,  must  be  one  inch. 

If  x=8,  then  y=4.  If  a:=30,  then  y=l5, 

Ifa;=10,         y  =  5,  Ifa:  =  100,         y=50,kc. 

On  the  other  hand,  if  i/  =  2,  then  x  =  4,  &c. 

529.  If  the  angle  HJlF  he  of  any  differen.  i.j.iagnitude,  as 
in  Fig.  18,  the  general  equation  will  be  the  same,  except  the 
co-efficient  of  x.  Let  the  ratio  of  ?/  to  a;  be  expressed  by  a, 
that  is,  let  y  :  x :  :  a  :  1.  Then  by  converting  this  into  an 
equation,  we  have 

ax=y. 
The  co-efficient  a  will  be  a  whole  number  or  a  fraction, 
according  as  y  is  greater  or  less  than  x. 

530.  To  apply  these  explanations  to  curves,  let  it  be  re- 
quired to  find  a  general  equation  of  the  common  parabola. 
(Fig.  19.)  It  is  the  distinguishing  property  of  this  figure,  as 
will  be  shown  under  Conic  Sections,  that  the  abscissas 
are  proportioned  to  the  squares  of  their  ordinates.  Let  the 
ratio  of  the  square  of  any  one  ordinate  to  its  abscissa,  be 
expressed  by  a.  As  the  ratio  is  the  same,  between  the 
square  of  any  other  ordinate  of  the  parabola  and  its  abscissa, 
we  have  universally  y^  :  x::  a  :  1 ;  and  by  converting  this 
into  an  equation, 

ax  =  y'^» 

This  is  called  the  equation  of  the  curve.  The  important 
advantages  gained  by  this  general  expression,  are  owing  to 
this,  that  the  equation  is  equally  applicable  to  every  point  of 
the  curve.  Any  value  whatever  may  be  assigned  to  the  ab- 
scissa X,  provided  the  ordinate  y  is  considered  as  belonging 
to  the  same  point.  But,  while  x  and  y  vary  together,  the 
quantity  a  is  supposed  to  remain  constant. 

By  the  equation  of  the  parabola,  ax=y'^,  and  extracting  the 
root  of  both  sides,  (Art.  297.) 

y  =  \^ax.     If  a=2,  theni/=V2£.     And 
If  x=  4.5=J1B (Fig.l9.)  then  y=\/2  x4.o=\/9z=S=BD 
U  x=  8.   =J1B^  y  =  \/2xS  =  \/l6  =  4=B'n' 

li  x=\2.b=AB"       '      j/  =  VlXl2.5=V25  =  5=J5^'iy' 
If;r=18.   =AB"'  r/=V2Xl8   z=zA^^^z=:Q^B'"iy". 


EQUATIONS  OF  CURVES.  311 

531.  When  ordinates  are  drawn  on  both  sides  of  the  axis 
to  which  tliey  are  applied  ;  those  on  one  side  will  be  positive, 
while  those  on  the  other  side  will  be  negative.  Thus,  in  Fig. 
1 9,  if  the  ordinates  on  the  upper  side  of  ^iF  be  considered  posi- 
tive, those  on  the  under  side  will  be  negative.  (Art.  507.) 
The  abscissas  also  are  either  positive  or  negative,  according 
as  they  are  on  one  side  or  the  other  of  the  point  from  which 
they  are  measured.  Thus,  in  Fig.  20,  if  the  abscissas  on  the 
right,  ^B,  AB',  &c.  be  considered  positive,  those  on  the  left, 
r^C,  AC,  &c.  will  be  negative.  And  in  the  solution  of  a 
problem,  if  an  abscissa  or  an  ordinate  is  found  to  be  negative, 
it  must  be  set  off  on  tli^  side  of  the  axis  opposite  to  that  on 
which  the  values  are  positive. 

532.  In  the  preceding  instances,  the  straight  line  or  curve  to 
which  the  ordinates  and  abscissas  are  applied,  crosses  the 
axis,  in  the  point  where  it  is  intersected  by  the  other  axis. 
Thus  the  curve  (Fig.  19.)  and  the  straight  line  E'D'{¥ig. 
20.)  cross  the  axis  AF,  in  the  point  A,  where  it  is  cut  by  the 
axis  AG.  But  this  is  not  always  the  case.  The  abscissas  on 
the  axis  QF,  (Fig.  21.)  may  be  reckoned  from  the  line  GJ^. 

Let  X  represent  any  one  of  the  abscissas,  MB,  MB',  &c. 
and  y  the  corresponding  ordinate. 

hetz=AB,  h=MA. 

And  a=  the  ratio  of  BD  to  AB,  as  before. 

Then  azz=y,  (Art.  529.)  that  is,  2=^- 

But  by  the  figure,  AB=MB  -  MA,  i.  e.  z=x-b 

Making  the  two  equations  equal,  x-b=y. 

a 

Therefore  a;=^-}-^- 

a 

533.  In  investigating  the  properties  of  curves,  it  is  impor- 
tant to  be  able  to  distinguish  readily  the  cases  in  which  the 
abscissas  or  ordinates  are  positive,  from  those  in  which  they 
are  negative;  and  to  determine  under  what  circumstances, 
either  of  the  co-ordinates  vanishes.  An  abscissa  vanishes  at 
the  point  where  the  curve  meets  the  axis  from  which  the  abscissas 
are  measured.  And  an  ordinate  vanishes,  at  the  point  where 
the  curve  meets  the  axis  from  which  the  ordmates  are 
measured. 


312  ALGEBUA. 

Thus,  ill  Fig.  19,  the  ordinates  are  measured  from  thehne 
AF.  TJie  length  of  eacli  ordinate  is  the  distance  of  a  particu- 
lar point  in  the  curve  from  the  hue.  As  the  curve  approaches 
the  axis,  the  ordinate  diminishes,  till  it  hecomes  nothing,  at 
tlie  point  of  intersection.  For,  here,  there  is  no  distance 
hetween  the  curve  and  the  axis. 

The  abscissas  are  measured  from  the  line  w3G.  These 
must  diminish  also,  as  the  curve  approaches  this  line,  and 
become  nothing  at  A. 

534.  From  tliis  it  is  evident,  that  when  the  two  axes  meet 
the  curve  at  the  same  pointy  the  two  co-ordinates  vanish  to- 
gether. In  Fig.  19,  the  two  axes  meet  the  curve  at  A,  the 
one  cutting,  and  the  other  touching  it.  But  in  Fig.  21,  the 
axis  J\IF  crosses  the  line  J\*D  at  A ;  while  GJ^  crosses  it  at 
JV.  The  ordinate,  being  the  distance  from  J\1F,  vanishes  at 
A,  where  the  distance  is  nothing.  But  the  abscissa,  being 
the  distance  from  GJ\'*,  vanishes  at  JV  or  M. 

535.  An  abscissa  or  an  ordinate  changes  from  positive  to 
negative,  b}^  passing  through  the  point  where  it  is  equal  to  0. 
Thus  the  ordinate  y,  (Fig.  20.)  diminishes  as  it  approaches 
the  point  A  ;  here  it  is  nothing,  and  on  the  other  side  of  A, 
it  becomes  negative,  because  it  is  below  the  axis  CF.  (Art. 
507.)  In  the  same  manner  the  abscissa,  on  the  right  of  AG, 
diminishes,  as  it  approaches  this  line,  becomes  0  at  A,  and 
then  negative  on  the  left. 

In  this  case,  the  two  co-ordinates  change  from  positive  to 
negative,  at  the  same  point.  But  in  Fig.  21,  the  ordinates 
change  from  positive  to  negative  at  A  ;  while  the  abscissas 
continue  positive  to  GJV,  being  still  on  the  riglit  of  that  line. 
On  the  right  from  A,  the  co-ordinates  are  both  positive  :  be- 
tween A  and  the  line  G^V,  the  abscissas  are  positive  :  and 
the  ordinates  negative:   and,  on  the  left  of  GJST  both  are 


536.  The  most  important  applications  of  the  principles 
stated  in  this  section,  will  come  under  consideration,  in  suc- 
ceeding branches  of  tlie  matlicmatics,  particidarly  in  Flux- 
ions. A  few  examples  ^\  ill  be  here  given  to  illustrate  the 
observations  which  liave  now  been  made. 

Prob.  1.  To  lind  the  equation  of  the  circle. 

In  the  circle  FGM,  (Fig.  22,)  let  the  two  diameters  OJV 
and  FJ\{  be  perpendicular  to  each  other.     From  any  point 


EQUATIONS  OF  CURVES.  31S 

in  the  curve,  draw  the  ordinate  DB  perpendicular  to  ^F; 
and  AB  will  be  the  corresponding  abscissa. 

Let  the  radius  AD=r,         AB=x,         BD=y. 

Then,  by  Euc.  47.  1,*  BD=:AD^AB 

That  is,  y''=r'^x' 


And  by  evolution,  r/= i^r' 


In  the  same  manner,  x=zt\/r^  ~  y^. 

That  is,  the  abscissa  is  equal  to  the  square  root  of  the  dif- 
ference between  the  square  of  the  radius  and  the  square  of 
the  ordinate. 

If  the  radius  of  the  circle  be  taken  for  a  unit,  (Art.  510)  its 
square  will  also  be  1 ,  and  the  two  last  equations  will  become 

=t\/\-x',  and  x=t^^\-'^f. 
These  equations  will  be  the  same,  in  whatever  part  of  the 
arc  GDF  the  point  D  is  taken.     For  the  co-ordinates  will  be 
the  legs  of  a  right  angled  triangle,  the  hypothenuse  of  which 
will  be  equal  to  AD,  because  it  is  the  radius  of  the  circle. 

537.  To  understand  the  application  to  the  other  quarters 
of  the  circle,  it  must  be  observed,  that,  in  each  of  the 
equations,  the  root  is  ambiguous.  The  values  of  y  and  of  x 
may  be  either  positive  or  negative.  This  results  from  the 
nature  of  a  quadratic  equation.  (Art.  297.)  It  corresponds 
also  with  the  situation  of  the  different  parts  of  the  circle,  with 
respect  to  the  two  diameters  FM  and  GJ^.  In  the  first 
quarter  GF,  the  co-ordinates  are  supposed  to  be  both  positive. 
In  the  second,  GM,  the  ordinates  are  still  positive,  but  the 
abscissas  become  negative.  (Art.  531.)  In  the  third,  JI/JV, 
both  are  negative,  and  in  the  fourth,  J^F,  the  ordinates  are 
negative,  but  the  abscissas  positive.     That  is, 

f   FG,  X  is  -f,  and  t/4-, 

In  the  quadrant  <!  ^^;;:[^' ^      :>         ^+^ 

*  Legendre,  186. 
40 


314  ALGEBRA. 

538.  In  geometry,  lines  are  supposed  to  be  produced  by 
the  motion  of  a  point.  If  the  point  moves  uniformly  in  one 
direction,  it  produces  a  straight  line.  If  it  continually  varies 
its  direction,  it  produces  a  curve.  The  particular  nature  of 
the  curve  depends  on  certain  conditions  by  which  the  motion 
is  regulated.  If,  for  instance,  one  point  moves  in  such  a 
manner,  as  to  keep  constantly  at  the  same  distance  from 
another  point  which  is  fixed,  the  figure  described  is  a  circle, 
of  which  the  fixed  point  is  the  centre.  It  is  evident  from 
the  preceding  problem,  that  the  equation  of  this  curve  de- 
pends on  the  manner  of  description.  For  it  is  derived  from 
the  property  that  different  parte*  of  the  periphery  are  equally 
distant  from  the  center.  In  a  similar  manner,  the  equations 
of  other  curves  may  be  derived  from  the  law  by  which  they 
are  described  ;  as  will  be  seen  in  the  following  examples. 

Prob.  2.  To  find  the  equation  of  the  curve  called  the  Cis- 
said  of  Diodes.      (Fig.  23.) 

The  description,  which  may  be  considered  as  the  clejinition 
of  the  figure,  is  as  follows. 

In  the  diameter  AB,  of  the  semi-circle  AJ^B,  let  the  point 
R  be  at  the  same  distance  from  B,  as  P  is  from  A.  Draw 
liJ^  perpendicular  to  AB,  to  cut  the  circle  in  JV".  From  v3, 
tluough  vV,  draw  a  straight  line,  extending  if  necessary  be- 
yond the  circle.  And  from  P,  raise  a  perpendicular,  to  cut 
this  line  in  J\I.     The  curve  passes  through  the  point  J>/. 

By  taking  P  at  different  distances  from  A,  as  in  Fig.  24, 
any  number  of  points  in  the  curve  may  be  determined.  As 
the  line  PJ\I  mo\es  towards  B,  it  becomes  longer  and  longer ; 
so  as  to  extend  the  Cissoid  beyond  the  semi-circle. 

To  find  the  equation  of  the  curve,  let  AH  and  JlB  be  the 
axes  of  the  co-ordniates. 

Also,  let  each  of  the  abscissas  AP,  AP'  AP",  &c.  =x, 
each  of  the  ordinates  PM,  FJf,  P"  M",  &c.  ^y, 
and  the  diameter       AB  =b. 

Then  by  the  construction,  PB—AB  -AP=  h  -  x. 

As  P.l/and  RJ^  are  each  perpendicular  to  AB,  the  trian- 
gles APM  and  ARJ^  are  similar.  (Euc.  27  and  29.  1.) 
Therefore, 


EQUATIONS  OF  CURVES.  315 

1.  By  similar  triangles,  AP  :  PM:  :  AR  :  RJ^T 

2.  Or,  by  putting  PB  for  its  equal  .QR, 

AP:  PJI::PB:RJ^ 

3.  Therefore,  P.l/xP^,^^^ 


4.  Squaring  both  sides,  ^'^^  ^  /^^  =~Rjy 

5.  By  Euc.  35.  3,  and  3.  3,*  .iRxRB=RJsr 

6.  Or,  putting  PB  for  its  equal  .IR,  and  Jlf  for  its  equal  /2^, 

PBx^p=Rjy 


7.  Making 4th  and  Gthequal,  PBx^iP=?^^~-^ 

ItP 


8.  Therefore,  .c^P  =PJ\f  xPB 

9.  Or,  x^=y^X(ft-^). 

That  is,  the  cube  of  the  abscissa  is  equal  to  the  square  of 
the  ordinate,  multipHed  by  the  difference  between  the  diame- 
ter of  the  circle,  and  the  abscissa.  The  equation  is  the  same 
for  every  pair  of  co-ordinates. 

Prob.  3.  To  find  the  equation  of  the  Conchoid  of  Nico- 
medes. 

To  describe  the  curve,  let  AB,  Fig.  25,  be  a  line  given  in 
position,  and  C  a  point  v/ithout  the  line.  About  this  point,  let 
the  line  Ch  revolve.  From  its  intersections  with  AB,  make 
the  distances  EM,  E'M,  E"M",  &c.  each  equal  to  AD. 
The  curve  will  pass  through  the  points  Z>,  M,  J\T,  M",  &c. 

To  find  its  equation,  let  CD  and  AB  be  the  axes  of  the  co- 
ordinates. Draw  FM  parallel  to  AP,  and  PM  parallel  to  CF. 
From  the  construction,  AD  is  equal  to  EM. 

Let  the  abscissa  AP=FM=x, 

the  ordinate  PM=AF=ij, 

the  given  hue      '  CA=ay 

and  AD=EM=b, 

Then  will  CF=  CA-\-AF=  a-f-i/. 

*  Legendra,  105,  224. 


316  ALGEBRA. 

As  CM  cuts  the  parallels  CD  and  PM,  and  also  the  paral- 
lels AP  and  FM,  the  triangles  CFM  and  MPE  are  similar. 
Then 

1.  By  similar  triangles,  CF  :  FM: :  PM  :  PE 

2.  Therefore,  PE=?^^<J^^ 

CF 


3.  Squaring  both  sides,       PE  = ~- 


CF 

2 


4.  By  Euc.  47.  1.  PE  =EM  -  PM 


_2 


imr-pM'^F^r  xPM 


5.  Mak.  3d  and  4th  equal,  EM  -  PM 

CF 


V2 


6.  That  is,  V-y'=J^ 

7.  Or,  (a+yYx{V-v')=^y'. 
539.  In  these  examples,  the  equation  is  derived  from  the 

description  of  the  curve.  But  this  order  may  be  reversed. 
If  the  equation  is  given,  the  curve  may  be  described.  For 
the  equation  expresses  the  relation  of  every  abscissa  to  the 
corresponding  ordinate.  The  curve  is  described,  therefore, 
by  taking  abscissas  of  different  lengths,  and  applying  ordinates  to 
each.  The  line  required,  will  pass  through  the  extremities  of 
these  ordinates. 

Prob.  4.  To  describe  the  curve  whose  equation  is 
2x=y\  or  y=z\/2x. 

On  the  line  AF,  (Fig.   19.)  take  abscissas  of  differeot 
lengths: 

For  instance,  AB=A.6,  tlien  the  ordinate  J5D=3,  (Art.  530.) 

AB'  =8.  B'D'  =  4, 

AB''=n,5  B'D' =5, 

AB"'=\^.  B'"D"'=Q, 

&c. 


EQUATIONS   OF   CURVES.  317 

Apply  these  several  ordiiiates  to  their  abscissas,  and  con- 
nect the  extremities  by  the  line  ADD^D^\  &c.  which  will  be 
the  ciirv^e  required.  The  description  will  be  more  or  less 
accurate,  according  to  the  number  of  points  for  which  ordi- 
nates  are  found. 

540.  If  a  point  is  conceived  to  move  in  such  a  manner,  as 
to  pass  through  the  extremities  of  all  the  ordinates  assigned 
by  an  equation  ;  the  line  which  it  describes  is  called  theTocw.v 
of  the  point,  that  is  the  path  in  which  it  moves,  and  in  which 
it  may  always  be  found.  The  line  is  also  called  the  locus  of 
the  equation  by  which  the  successive  positions  of  the  point  ere 
determined.  Thus  the  common  parabola  (Fig.  19,)  is  called 
the  locus  of  the  points,  D,  D',  D'',  Slc.  or  of  tJie  equation 
ax=y\  (Art.  530.)  Tiie  arc  of  a  circle  is  the  locu»  of  the 
equation  x=±\/r-y\  (Art  536.)  To  find  the  locus  oU 
an  equation,  therefore,  is  the  same  thing,  as  to  find  the 
straight  line  or  curve  to  which  the  equation  belongs. 

Prob.  5.  To  find  the  locus  of  the  equation 

x=y,  or  ax=:y, 

a 

in  whicli  x  and  y  are  variable  co-ordinates,  while  a  is  a  deter- 
minate quantity. 

If  the  abscissa  x  be  taken  of  different  lengths,  the  ordinate 
y  must  vary  in  such  a  manner  as  to  preserve  ax=y  ;  or  con- 
verting the  equation  into  a  proportion,  y  :  x  :  :  a  :  1.  There- 
fore, as  a  is  a  determinate  quantity,  the  ratio  of  x  to  y  will  be 
invariable  ;  that  is,  any  one  abscissa  will  be  to  its  ordinate  as 
any  other  abscissa  to  its  ordinate.  Let  two  of  the  abscissas 
be  ,^B  and  AB',  (Fig.  17.)  and  their  ordinates,  BD  and 
B'D;  then, 

AB  '.BD::  AB'  :  B'U. 

The  line  ADD'  is,  therefore,  a  straight  line ;  (Euc.  32.  6.) 
and  this  is  the  locus  of  the  equation. 

If  the  proposed  equation  is  x=z±-\-h,  the  additional  term  b 

a 

makes  no  difference  in  the  nature  of  the  locus.  For  the  only 
effect  of  6,  is  to  lengthen  the  abscissas,  so  that  they  must  not 
be  measured  from  A,  but  from  some   other  point,  as  M, 


318  ALGEBRA. 

(Fig.  21.)  The  ratio  of  JlB,AB',  &c.  to  BD',B'iy,  &c.  still 
remains  the  same.  See  Art.  532.  The  locus  of  the  equation 
is,  therefore,  a  straight  line. 

541.  From  this  it  will  be  easy  to  prove,  that  the  locus  of 
every  equation  in  which  the  co-ordinates  x  and  y  are  in  sepa- 
rate terms,  and  do  not  rise  above  the  first  power,  is  a  straight 
line.     For  every  such  equation  may  be  brought  to  the  form 

7:=y.tb.     All  the  terms  may  be  reduced  to  three,  one  con- 
a 

taining  a:,  another  y,  and  a  third,  the  aggregate  of  the  con- 
stant quantities  which  are  not  co-efficients  of  :c  and  y  ;  as  will 
be  seen  in  the  following  problem. 

Prob.  6.  To  find  the  locus  of  the  equation 

ex  -  d-\-hx  -  y-\-in=n. 
By  transposition,  cx-\-hx=y-\-n  -  m~\-d. 

Dividing  by  c-\-h  x—-J. — -j-— ^^^ — il . 

c-j-/i         c-{-h 

Here  the  constant  quantities,  in  each  term,  may  be  repre- 
sented by  a  single  letter.    (Art.  321.)    !f,   then,  we  make 

c-{-h=n,  and  ^^  ~  ^^^~r   _  5 .  the  equationwill  become  a: =i/-|-^» 
c-\-h  a 

whose  locus,  by  the  last  article,  is  a  straight  line. 

542.  But  if  the  ordinates  are  as  the  squares,  cubes,  or 
higher  powers  of  the  abscissas,  the  locus  of  the  equation,  in- 
stead of  being  a  straight  line,  is  a  curve.  For  the  ordinates 
applied  to  a  straight  line,  have  the  same  ratio  to  each  other 
which  their  abscissas  have.  But  quantities  have  not  the 
same  ratio  to  each  other,  which  their  squares,  cubes,  or  higher 
powers  have.  (Art.  354.)  Thus,  if  x'^=y,  the  ordinates 
will  increase  more  rapidly  than  the  abscissas.  If  the  abscis- 
sas be  taken,  1,  2,  3,  4,  &c.  the  ordinates  will  be  equal  to 
their  squares,  1,  4,  9,  16,  &c, 

543.  As  an  unlimited  variety  of  equations  may  be  produ- 
ced, by  different  combinations  and  powers  of  the  co-ordi- 
nates, and  as  each  of  these  has  its  appropriate  locus  ;  it  is 
evident  that  the  forms  of  curves  must  be  inniunerable.  The}' 
may,  however,  be  reduced  to  classes.  The  modern  mode  of 
classing  them,  is  from  the  degree  of  their  equations.      The 


EQUATIONS  OF  CURVES.  319 

different  orders  of  lines  are  dislinguished,  by  the  greatest  index, 
or  sum  of  the  indices  of  the  co-ordinates^  in  any  term  of  the 
equation. 

Thus  tlie^equatioii  ax=y  belongs  to  a  line  of  the  first  or- 
der, because  the  index  of  each  of  the  co-oidinates  is  1.  But 
this  order  includes  no  curves.  For,  by  Art.  541,  the  locus  of 
every  such  equation  is  a  straight  hne. 

The  equation  cx"^  -  axy  =:y^,  belongs  to  the  second  order  of 
lines,  or  the  first  kind  of  curves,  because  the  greatest  index 
is  2.  The  equation  ay-\-xy  =  bx  also  belongs  to  the  second 
order.  For,  althougli  there  is  here  no  index  greater  than 
1,  yet  the  sum  of  the  indices  of  x  and  y,  in  the  second  term, 
is  2. 

The  equation  1/^-3 aa:7/  =  62-"  belongs  io  the  third  order  of 
lines,  or  the  second  kind  of  curves,  because  the  greatest  in- 
dex of  y  is  3. 

544.  In  curves  of  the  higher  orders,  the  ordinate  belong- 
ing to  any  given  abscissa  may  have  different  values,  and  may 
therefore  meet  the  curve  in  several  points.  For  the  length 
of  the  ordinate  is  determined  by  the  equation  of  the  curve, 
and  if  the  equation  is  above  the  first  degree,  it  may  have  two 
or  more  roots,  (Art.  498.)  and  ma}^  therefore,  give  different 
values  to  the  ordinate. 

An  equation  of  the  first  degree  has  but  one  root ;  and  a 
hne  of  the  first  order,  can  be  intersected  by  an  ordinate,  in 
one  point  only.  Thus  the  equation  of  the  line  AH  (Fig. 
17.)  is  ax=y,  in  which  it  is  evident  y  has  but  one  value, 
while  X  remains  the  same.  If  the  abscissa  x  be  taken  equal 
to  AB,  the  ordinate  y  will  be  BD,  which  can  meet  the  line 
AH  in  D  only. 

But  the  equation  of  the  parabola  ?/"=r:«a:,  (Art.  530.)  has 
two  roots.  For,  by  extracting  both  sides,  y:=t\/ax.  (Art, 
297.)  It  is  true,  that  in  this  case,  the  two  values  of  y  are 
equal.  But  one  is  positive,  and  the  other  negative.  This 
shows  that  the  ordinate  may  extend  both  ways  from  the  end 
of  the  abscissa,  and  may  meet  the  opposite  branches  of  the 
curve.  Thus  the  ordinate  of  the  abscissa  AB  (Fig.  19.)  may 
be  either  BD  above  the  abscissa,  or  Bd  below  it. 

A  cubic  equation  has  three  roots  ;  and  an  ordinate  of  the 
curve  belonging  to  this  equation,  may  liave  three  different 
values,  and  may  meet  the  curve  in  three  different  points. 
Thus  the  ordinate  of  the  abscissa -^^  (Fig.  26.)  may  be  BD^ 
or  BD,  or  Bd. 


320  ALGEBRA. 

545.  When  the  curve  meets  the  axis  on  which  tlie  abscis- 
sas are  measured,  the  ordinate,  after  becoming  less  and  less, 
is  reduced  to  nothing-.  (Art.  533.)  But,  in  some  cases,  a 
curve  may  continually  approach  a  line,  without  over  meeting 
It.  Let  the  distances  JIB,  BB\  B'B",  &c.  on  the  line  AF, 
(Fig.  27.)  be  equal;  and  let  the  curve  DD'Z)^  &c.  be  of 
such  a  nature  that  of  the  severed  ordinates  at  the  points  B,B\ 
B",  &c.  each  succeeding  one  shall  be  half  the  preceding, 
that  is,  B'B',  half  BB,  B"B"  half  B'B',  &c.  It  is  evident 
that,  however  far  the  straight  hne  be  carried,  the  curve  will 
become  nearer  and  nearer  to  it,  and  yet  will  never  quite  reach 
it.  A  line  which  thus  continwally  approaches  a  curve  without  ever 
meeting  it,  is  called  an  asymptote  of  the  curve.  The  axis  AF 
is  here  the  asymptote  of  the  curve  BB'B",  &c.  As  the  ab- 
scissa increases,  the  ordinate  diminishes,  so  that,  when  the 
abscissa  is  mathematically  infinite,  (Art.  447.)  the  ordinate 
becomes  an  infinitesimal,  and  may  be  expressed  by  0.  (Art. 
455.)* 

*  See  Note  Y. 


NOTES. 


Note  A.  Page  1. 

As  the  term  quantity  is  here  used  to  signify  whatever  is 
the  object  of  mathematical  inquiry,  it  will  be  obvious  that 
number  is  meant  to  be  included  ;  so  far  at  least,  as  it  can  be 
the  subject  of  mathematical  investigation.  Dugald  Stewart 
asserts,  indeed,  that  it  might  be  easily  shown,  that  number 
does  not  fall  under  the  definition  of  quantity  in  any  sense  of 
that  word.  Philosopby  of  the  Mind,  Vol.  II.  Note  G.  For 
proof  that  it  is  included  in  the  common  acceptation  of  the 
word,  it  will  be  sufficient  to  refer  to  almost  any  mathematical 
work  in  which  the  term  quantity  is  explained,  and  particu- 
larly to  the  familiar  distinction  between  continued  quantity  or 
magnitude,  and  discrete  quantity  or  number. 

But  does  number  "fall  under  the  definition  of  quantity  ?" 
Mr.  Stewart  after  quoting  the  observation  of  Dr.  Reid,  that 
the  object  of  the  mathematics  is  commonly  said  to  be  quan- 
tity, which  ought  to  be  defined,  that  ivhicfi  may  be  measured, 
adds,  "  The  appropriate  objects  of  this  science  are  such 
things  alone  as  admit  not  only  of  being  increased  and  dimin- 
ished, but  of  being  midtipUed  and  divided.  In  other  words, 
the  common  character  which  characterizes  all  of  them,  is 
their  mensurability.^^  That  number  may  be  multiplied  and 
divided,  will  not  probably  be  questioned.  But  it  may  per- 
haps be  doubted,  whether  it  is  capable  of  mensuration.  If, 
as  Mr.  Locke  observes,  "number  is  that  which  the  mind 
makes  use  of,  in  measuring  all  tilings  that  are  measurable,'' 
can  it  measure  itself,  or  be  measured  ]  It  is  evident  that  it  can 
not  be  measured  geometrically,  by  applying  to  it  a  measure  of 
length  or  capacity.  But  by  measuring  a  quantity  mathe- 
matically, what  else  is  meant,  than  determining  the  ratio 
which  it  bears  to  some  other  quantity  of  the  same  kind  ;  in 
other  words  finding  how  often  one  is  contained  in  the  other, 
either  exactly  or  with  a  certain  excess  1  And  is  not  this  as 
applicable  to  number  as  to  magnitude  1    The  ratio  which  a. 

41 


322  ALGEBRA. 

given  nvimber  bears  to  v.nify  cannot,  indeed,  be  the  subject 
of  inquiry ;  because  it  is  expressed  by  the  number  itself. 
But  the  ratio  which  it  bears  to  other  numbers  may  be  as  pro- 
per an  object  of  mathematical  investigation,  as  the  ratio  of  a 
mile  to  a  furlong. 

For  proof  tlia't  number  is  not  quantity,  Mr.  Stewart  refers 
to  Barrow's  Mathematical  Lectures.  Dr.  Barrow  has  start- 
ed an  etymological  objection  to  the  application  of  the  term 
quantity  to  number,  which  he  intimates  might,  with  more 
propriety,  be  called  quotity.  He  observes,  "  The  general  ob- 
ject of  the  mathematics  has  no  proper  name,  either  in  Greek 
or  Latin."  And  adds,  "  It  is  plain  the  mathematics  is  con- 
versant about  two  things  especially,  quantity  strictly  taken, 
and  quotity  ;  or  magnitude  and  multitude."  There  is  fre- 
quent occasion  for  a  common  name,  to  express  number,  dura- 
tion, &c.  as  well  as  magnitude  ;  and  the  term  quantity  will 
probably  be  used  for  this  purpose,  till  some  other  word  is  sub- 
stituted in  its  stead. 

But  though  Dr.  Barrow  tlius  distinguishes  between  mag- 
nitude and  number,  he  afterwards  gives  it  as  his  opinion, 
(page  20,  49,)  that  there  is  really  no  quantity  in  nature  dif- 
ferent from  what  is  called  magnitude  or  continued  quantity, 
and  consequently,  that  this  alone  ought  to  be  accounted  the 
object  of  the  mathematics.  He  accordingly  devotes  a  whole  lec- 
ture to  the  purpose  of  proving  the  identity  of '  arithmetic  and 
geometry.  {Led.  3.)  He  is  "convinced  that  number  really 
differs  nothing  from  what  is  called  continued  quantity  ;  but 
is  only  formed  to  express  and  declare  it ;"  that  as  "  the  con- 
ceptions of  magnitude  and  number  could  scarcely  be  separa- 
ted," by  the  ancients,  "  in  the  name,  they  can  hardly  be  so 
in  the  7nmf/,"  and  "  that  number  includes  in  it  every  conside- 
ration pertaining  to  geometry."  He  admits  of  metaphysical 
number,  which  is  not  the  object  of  geometry,  or  even  of  the 
mathematics.  But,  in  his  view,  magnitude  is  always  inclu- 
ded in  mathematical  number,  as  the  units  of  which  it  is  com- 
posed are  equal.  On  the  other  hand,  magnitudes  are  not 
to  be  considered  as  mathematical  quantities,  except  as  they 
are  measured  by  number.  In  short,  quantity  is  magnitude 
measured  by  number. 

It  would  seem,  then,  that  according  to  Dr.  Barrow,  num- 
ber considered  as  separate  from  magnitiKle,  has  as  fair  a 
claim  to  be  called  quantity,  as  magnitude  considered  as  sep- 
arate from  number.      If   arithmetic  and  geometry  are  the 


NOTES.  323 

same;  quantity  is  as  much  the  object  of  one,  as  of  the  other. 
How  far  this  scheme  is  apphcable  to  duration,  motion,  &c.  it 
is  nx)t  necessary,  in  this  place  to  inquire. 

Note  B.  p.  1. 

It  is  to  be  regretted,  that  the  science  of  Fluxions  has  re- 
ceived its  name  from  the  particular  manner  in  which  its  in- 
ventor, Sir  Isaac  Newton,  explained  its  principles,  rather  than 
from  the  nature  of  the  science  itself.  This  has  served  to 
countenance  the  opinion,  that  th-e  doctrine  of  fluxions,  and 
the  differential  and  integral  calculus,  in  which  a  different  laji- 
guage,  and  different  mode  of  explanation  have  been  adopted, 
are  distinct  methods  of  investigation.  Whereas  the  funda- 
mental laws  of  calculation  are  the  same  in  both.  These 
have  no  necessary  dependence  on  motion,  or  even  on  geo- 
metrical magnitudes.  The  method  of  fluxions  has  been 
greatly  enlarged  and  modified  since  Newton's  day.  But  it 
is  difficult  to  change  the  name,  to  adapt  it  to  the  present 
state  of  the  science,  without  seeming  to  derogate  from  that 
profound  regard  which  is  due  to  tlie  original  inventor. 

Note  C.  p.  32. 

It  is  common  to  define  multiplication,  by  saying  that  *  it  is 
finding  a  product  which  has  the  same  ratio  to  the  multipli- 
cand, that  the  multiplier  has  to  a  unit.'  This  is  strictly  and 
universally  true.  But  the  objection  to  it,  as  a  definition,  is, 
that  the  idea  of  ratio,  as  the  term  is  understood  in  arithmetic 
and  algebra,  seems  to  imply  a  previous  knowledge  of  multi- 
plication, as  well  as  of  division.  In  this  Avork  at  least,  the 
expression  of  geometrical  ratio  is  made  to  depend  on  division, 
and  division  on  multiplication.  Ratio,  therefore,  could  not, 
be  properly  introduced  into  the  definition  of  midtiplication. 

It  is  thought,  by  some,  to  be  absurd  to  speak  of  a  unit  as 
consisting  of  parts.  But  w^hatever  may  be  true  with  respect 
to  number  in  the  abstract,  there  is  certainly  no  absurdity  in 
considering  an  integer,  of  one  denomination,  as  made  up  of 
parts  of  a  different  denomination.  One  rod  may  contain 
several  feet :  one  foot  several  inches,  &c.  And  in  multipli- 
cation, we  may  be  required  to  repeat  the  whole,  or  a  part  of 
the  multiplicand,  as  many  times  as  there  aie  inches  in  a  foot, 
or  part  of  a  foot 


324  ALGEBRA. 

Note  D.  p.  66. 

It  is  perhaps  more  philosophically  exact,  to  consider  an 
equation  as  affirming  the  equivalence  of  two  different  expres- 
sions of  the  same  quantity,  than  to  speak  of  it  as  expressing 
an  equality  between  one  quantity  and  another.  But  it  is 
doubted  whether  the  former  definition  is  the  best  adapted  to 
the  apprehension  of  the  learner;  who  in  this  early  part  of  his 
mathematical  course,  may  be  supposed  to  be  very  little  accus- 
tomed to  abstraction.  Though  he  may  see  clearly,  that  the 
area  of  a  triangle  is  equal  to  the  area  of  a  parallelogram  of 
the  same  base  and  half  the  height ;  yet  he  may  hesitate  in 
pronouncing  that  the  two  surfaces  are  precisely  the  same. 

Note  E.  p.  86. 

As  the  direct  powers  of  an  integral  quantity  have  positive 
indices,  while  the  reciprocal  powers  have  negative  indices  ;  it 
is  common  to  call  the  former  positive  poit'ers,  and  the  latter 
negative  powers.  But  this  language  is  ambiguous,  and  may 
lead  to  mistake.  For  the  same  terms  are  applied  to  powers 
with  positive  and  Jiegative  signs  prefixed.  Thus  -|-8a^  is 
called  a  positive  power  ;  while  -  Sa*  is  called  a  negative  one. 
It  may  occasion  perplexity,  to  speak  of  the  latter  as  being 
both  positive  and  negative  at  the  same  time ;  positiv^e,  be- 
cause it  has  a  positive  index^  and  negative  because  it  has  a 
negative  co-efficient.  This  ambiguity  may  be  avoided,  by 
using  the  terms  direct  and  reciprocal ;  meaning,  by  the  for- 
mer, powers  with  positive  exponents,  and  by  the  latter,  pow- 
ers with  negative  exponents. 

Note  F.  p.  109. 

I  liave  been  unwilling  to  admit  into  the  text  the  rules  of 
calculation  which  are  conmionly  applied  to  imaginary  quan- 
tities ;  as  mathematicians  have  not  yet  settled  the  logic  of 
the  principles  upon  which  these  rules  must  be  founded.  It 
appears  to  be  taken  for  granted  by  Euler  and  others,  that  the 
product  of  the  imaginary  roots  of  two  quantities,  is  equal  to 
the  root  of  the  product  of  the  quantities ;  for  instance,  that 

V-axV-6=:V-aX-^^-     I^^  ^^^is  principle  be  admitted, 
certain  limitations  must  be  observed  in  the  application.     If 

we  make  V-axV-a  =  V-aX-«j  and  this  in  confor- 
mity with  the  common  rule  for  possible  quantities,  =/sJa^ ) 


NOTES.  325 

yet  we  are  not  at  liberty  to  consider  the  latter  expression  as 
equivalent  to  a.  For  though  \fd',  when  taken  without  re- 
ference to  its  origin,  is  ambiguous,  and  may  be  cither  -\-a  or 
-a;  yet  when  w^e  know  that  it  has  been  produced  by  mul- 
tiplyingV^^  into  itself,  w^e  are  not  permitted  to  give  it  any 
other  value  than  -  a.  (Art.  262.) 

On  the  principle  here  stated,  imaginary  expressions  may 
be  easily  prepared  for  calculation,  by  resolving  the  quantity 
under  the  radical  sign  into  two  factors,  one  of  which  is  -\  \ 
thereby  reducing  the  imaginary  part  of  the  expression  to  V-1. 
As  -a:=-j-aX  -  1?  the  expression  \/  -a=i^ax  -  1=\/^X 

V^.  So  V-a-Z^^Vrt-f^xV^.  The  first  of  the 
two  factors  is  a  real  quantity.  After  the  impossible  part  of 
imaginary  expressions  is  thus  reduced  to  V-  1,  they  may  be 
multiplied  and  didded  by  the  rules  already  given  for  other 
radicals. 

Thus  in  Multiplication, 

1.  V^<xV^=V«xV^xv^xV^r=^a6x-i  = 

2'.  +V-"ax  -  V~6=  - \/ah X  -  1  =^-\fah. 
3.  V~9xV^^-V36=-6. 

4.  (i+v~r)x(i-v~r)=2. 

From  these  examples  it  will  be  seen,  that  according  to  the 
principle  assumed,  the  product  of  two  imaginary  expressions 
is  a  real  quantity. 

5.  V~rtxV^=V«xV^xV^  =  V«^xV^. 

6.  V~2xVl8  =  6xV-T. 

Hence,  the  product  of  a  real  quantity  and  an  imaginary 
expression,  is  itself  imaginary. 

In  Division,  

1    V~t _ V^ X V^_      /«  2.  ^-^-1. 

'  V^    V^xV^    ^  ^'        "'  V^ 

Hence,  the  quotient  of  one  imaginary  expression  divided 
by  another  is  a  real  quantity. 

Q    V^_V^xV~i_     /^s^V^ 


326  ALGEBRA. 


4. 


/s/a  _         \/a 


V-«     v«xV-i     V-^ 


Hence,  the  quotient  of  an  imaginary  quantity  divided  by  a 
real  one,  or  of  a  real  quantity  divided  by  an  imaginary  one, 
is  itself  imaginary. 

By  multiplying  V-  I-continually  into  itself,  we  obtain  the 
following  powers. 

(V^)'=  - 1  _     (^ZL)'=  - 1 

(V-i)^=-V-r    {V-i_y=-V-i 
(V~iy=-{.l (V~i)^:^4-i  

(V-iy=+v-\    (V-i)^^+v-i 

&c.  &c. 

The  even  powers  being  alternately  -  1  and  +1  and  the 
odd  powers,  -  V  -  T  and  -{-"V  -  1 . 

On  the  nature  and  use  of  imaginary  expressions,  see  Eu- 
ler's  Algebra,  Rees'  Cyclopedia,  the  Edinburgh  Review,  Vol. 
I.  and  the  London  Philosophical  Transactions  for  1801, 1802 
and  1806. 

Note  G.  p.  146. 

Every  affected  quadratic  equation  may  be  reduced  to  one 
of  the  three  foUowinsr  forms. 


2.x'' 
S.x'' 


-\-ax=  h  ^ 
-ax=  '  b> 
-ax=  -b) 


These,  when  they  t^re  resolved,  become 


L  x=--ia±V^a'-\-b   ) 

2.  x=    Ja±VK-f6   > 

3.  x=    iatVia'-b   ) 


2 
3 

In  the  two  first  of  these  forms,  the  roots  are  never  imagi 
nary.     For  the  terms  under  the  radical  sign  are  both  posi 
tive.     But  in  the  third  form,  whenever  b  is  greater  than  Ja% 
the  expression  W  -b  is  negative,  and  therefore  its  root  is 
impossible. 


NOTES.  337 

Note  H.  p.  175. 

For  the  sake  of  keeping  clear  of  the  muhiphed  controver- 
sies, a  great  portion  of  them  verbal,  respecting  the  nature  of 
ratio,  I  have  chosen  to  define  geometrical  ratio  to  be  that 
which  is  expressed  by  the  quotient  of  one  quantity  divided  by 
another,  rather  than  to  say  that  it  consists  in  this  quotient. 
Every  ratio  which  can  be  mathematically  assigned,  may  be 
expressed  in  this  way,  if  we  include  surd  quantities  among 
those  which  are  to  be  admitted  into  the  numerator  or  denomi- 
nator of  the  fraction  representing  the  quotient. 

Note  I.  p.  177. 

This  definition  of  compound  ratio  is  more  comprehensive 
than  the  one  which  is  given  in  Euclid.  That  is  included  in 
this,  but  is  limited  to  a  particular  case,  which  is  stated  ia 
Art.  353.  It  may  answer  the  purposes  of  geometry,  but  is 
not  sufficiently  general  for  algebra. 

Note  K.  p.  178. 

It  is  not  denied  that  very  respectable  wi iters  use  these 
terms  indiscriminately.  But  it  appears  to  be  without  any 
necessity.  The  ratio  of  6  to  2  is  3.  There  is  certainly  a 
difference  between  twice  this  ratio,  and  the  square  of  it,  that 
is,  between  twice  three,  and  the  square  of  tlnee.  All  are 
agreed  to  call  the  latter  a  duplicate  ratio.  What  occasion  is 
there,  then,  to  apply  to  it  the  term  do^le  also  ]  Tliis  is 
wanted,  to  distinguish  the  other  ratio.  And  if  it  is  confined 
to  that,  it  is  used  according  to  the  common  acceptation  of  the 
word,  in  famihar  language.  J 

Note  L.  p.  185. 

The  definition  here  given  is  meant  to  be  applicable  to 
quantities  of  every  description.  The  subject  of  proportion  as 
it  is  treated  of  in  Euclid,  is  embarrassed  by  the  meai.s  which 
are  taken  to  provide  for  the  case  of  incommensurable  quanti- 
ties. But  this  difficulty  is  avoided  by  the  algebraic  nota- 
tion which  may  represent  the  ratio  even  of  incommensur- 
ables. 

Thus  the  ratio  of  1  to  \/2  is 

V2 


328  ALGEBRA. 

It  is  impossible,  indeed,  to  express  in  rational  numbers, 
the  square  root  of  2,  or  the  ratio  which  it  bears  to  1.  But 
this  is  not  necessary,  for  the  purpose  of  showing  its  equality 
with  another  ratio. 

The  product  4x2  =  8. 

And,  as  equal  quantities  have  equal  roots, 

2XV2  =  V^'  therefore,  2  :  V^  -  1  ^  V^- 
Here  the  ratio  of  2  to  a^S,  is  proved  to  be  the  same,  as 
that  of  1  to  V^  5  although  we  are  unable  to  find  the  exact 
value  either  of  \/8  or  \/2. 

It  is  impossible  to  determine,  with  perfect  accuracy,  the 
ratio  which  the  side  of  a  square  has  to  its  diagonal.  Yet  it 
is  eas)^  to  prove,  that  the  side  of  one  square  has  the  same  ra- 
tio to  its  diagonal,  which  the  side  of  any  other  square  has  to 
its  diagonal.  When  incommensurable  quantities  are  once 
reduced  to  a  proportion,  they  are  subject  to  the  same  laws  as 
other  proportionals.  Throughout  the  section  on  proportion, 
the  demonstrations  do  not  imply  that  we  know  the  vahie  of 
the  terms,  or  their  ratios  ;  but  only  that  one  of  the  ratios  is 
equal  to  the  other. 

Note  M.  p.  190. 

The  inversion  of  the  means  can  be  made  with  strict  pro- 
priety in  those  cases  oidy  in  which  all  the  terms  are  quanti- 
ties of  the  same  kind.  For,  if  the  two  last  be  different  from 
the  two  first,  the  antecedent  of  each  couplet,  after  the  inver- 
sion will  be  different  from  the  consequent,  and  therefore, 
there  can  be  no  ratio  between  them.   (Art.  355.) 

This  distinction,  however,  is  of  little  importance  in  prac- 
tice. For,  when  the  several  quantities  are  expressed  in  num- 
bers, there  will  always  be  a  ratio  between  the  numbers.  And 
when  two  of  them  are  to  be  multiplied  together,  it  is  imma- 
terial which  is  the  multiplier,  and  which  the  nniltiplicand. 
Thus  in  the  Rule  of  Three  in  arithmetic,  a  change  in  tlie 
order  of  the  two  middle  terms  will  make  no  difference  in  the 
resul  t. 

Note  N.  p.  197. 

The  terms  composition  and  division  are  derived  from  ge- 
ometry, and  are  introduced  here,  because  they  are  generally 
used  by  writers  on  i)roportion.   But  they  are  calculated  rather 


NOTES.  329 

to  perpiex,  than  to  assist  the  learner.  The  objection  to  the 
word  composition  is,  that  its  meaning  is  hable  to  be  mistaken 
for  the  composition  or  compounding  of  ratios.  (Art.  300.) 
The  two  cases  are  entirely  ditferent,  and  ought  to  be  carefully 
distinguished.  In  one,  the  terms  are  addecl^  in  tiie  other, 
they  are  multiplied  together.  The  word  compound  has  a  simi^^ 
lar  ambiguity  in  other  parts  of  the  mathematics.  The  ex- 
pression a-{-b,  in  which  a  is  added  to  b,  is  called  a  compound 
quantity.  The  fraction  J  of  -f,  or  J  xh  in  which  |  is  multi- 
plied into  f,  is  called  a  compound  fraction. 

The  term  division,  as  it  is  used  here,  is  also  exceptionable. 
The  alteration  to  which  it  is  applied,  is  effected  by  subtraction, 
and  has  nothing  of  the  nattire  of  what  is  called  division  in 
arithmetic  and  algebra.  But  there  is  another  case,  (Art. 
392.)  totally  distinct  from  this,  in  which  the  change  in  the 
terms  of  the  proportion  is  actually  produced  by  division. 

Note  O.  p.  206. 

The  principles  stated  in  this  section,  are  not  only  expressed 
in  different  language,  from  the  corresponding  propositions  in 
Euchd,  but  are  in  several  instances  more  general.  Thus  the 
first  proposition  in  the  fifth  book  of  the  Elements,  is  confined 
to  equimultiples.  But  the  article  referred  to,  as  containing  this 
proposition,  is  applicable  to  all  cases  of  equal  ratiosy  whether 
the  antecedents  are  multiples  of  the  consequents  or  not. 

Note  P.  p.  222. 

The  solution  of  one  of  the  cases  is  omitted  in  the  text,  be- 
cause it  is  performed  by  logarithms,  with  which  the  learner 
is  supposed  not  to  be  acquainted,  in  this  part  of  the  course. 
When  the  first  term,  the  last  term,  and  the  ratio  are  given, 
the  number  of  terms  may  be  found  by  the  fonnula 

log.-. 
**=  log.  r  * 

Note  Q.  p.  227. 

When  it  is  said  that  a  mathematical  quantity  may  be  sup- 
posed to  be  increased  beyond  any  determinate  limits,  it  is  not 
intended  that  a  quantity  can  be  specified  so  great,  that  no 
limits  greater  than  this  can  be  assigned.     The  quantity  and 

42 


330  ALGEBRA. 

the  limits  may  be  alternately  extended  one  beyond  the  other. 
If  a  Vme  be  conceived  to  rcacli  to  tlie  most  distant  point  in 
the  visible  heavens,  a  limit  may  be  mentioned  beyond  this. 
The  hnc  may  then  be  snpposed  to  be  extended  farther  than 
this  Hmit.  Another  point  may  be  specified  still  farther  on, 
and  yet  the  line  may  be  conceived  to  be  carried  beyond  it. 

Note  R.  p.  230. 

The  apparent  contradictions  respecting  infinity,  are  owing 
to  the  ambiguity  of  the  term.  It  is  often  thought  that  the 
proposition,  that  quantity  is  infinitely  divisible,  involves  an 
absurdity.  If  it  can  be  proved  that  a  line  an  imh  long  can 
be  divided  into  an  infinite  number  of  parts,  it  can,  by  the 
same  mode  of  reasoning,  be  proved,  that  a  line  two  inches 
long  may  be  first  divided  in  the  middle,  and  then  each  of  the 
sections  be  divided  into  an  infinite  number  of  parts.  In  this 
way,  we  shall  obtain  one  infinite  twice  as  great  as  another. 

If  by  infinity,  here  is  meant  that  which  is  beyond  any  as- 
signable limits,  one  of  these  infinites  may  be  supposed  gieater 
than  the  other,  without  any  absurdity.  But  if  it  be  meant 
that  the  number  of  divisions  is  so  great  that  it  cannot  be  in- 
creased, we  do  not  prove  this,  concerning  either  of  the  lines. 
We  make  out,  therefore  no  contradiction.  The  apparent 
ab3urdity..arise3  from  shifting  the  meaning  of  the  terms.  We 
demonstrate  that  a  quantity  is,  in  one  sense  infinite  ;  and 
then  infer  that  it  is  infinite,  in  a  sense  widely  diflferent. 

Note  S.  p.  233. 

Strictly  speaking,  the  inquiry  to  be  made  is,  how  often  the 
whole  divisor  is  contained  in  as  many  terms  of  the  dividend. 
But  it  is  easier  to  divide  by  a  part  only  of  the  divisor  ;  and 
this  Avill  lead  to  no  error  in  the  result,  as  the  whole  divisor  is 
multiplied,  in  obtaining  the  several  subtrahends. 

Note  T.  p.  244. 
The  demonstration  of  this  proposition,  particularly  in  its 
application  to  fractional  indices,  could  not  be  introduced,  with 
advantage,  in  tliis  part  of  the  course.  It  does  not  appear 
that  Newton  himself  demonstrated  his  theorem,  except  by 
induction.  And  thougli  various  demonstrations  have  since 
been  given  ;  yet  they  are  generally  founded  upon  principles 
and  methods  of  investigation  not  contained  in  this  introduc- 
tion, such  as  the  /aws  of  combination,  fluxions,  and  figurate 
numbers. 


NOTES.  331 

Those  who  wish  to  examine  the  inquiries  on  this  subject, 
may  consuU  Simpson's  Algebra,  Section  15,  Euler's  Algebra, 
Section  2,  Chap.  11,  Vince's  Fluxions,  Art.  99,  Lacroix'a 
Algebra,  Art.  138,  &c.  Do.  Comp.  Art.  71,  Rees'  Cyclopedia, 
Mamiing's  Algebra,  the  London  Phil.  Trans.  Vol.  xxxv,  p. 
298,  Woodhouse's  Analytical  Calculations,  Bonnycastle's 
Algebra,  and  Lagrange's  Theory  of  Analytical  Functions. 

Note  U.  p.  277. 

The  very  limited  exteiU  of  this  work  would  admit  of  no- 
thing more,  than  a  few  specimens  of  the  Summation  of  Se- 
ries." For  information  on  this  subject,  the  learner  is  referred 
to  Emerson's  Method  of  Increments,  Sterling's  Summation 
of  Series,  Waring's  Fluxions,  Maclaurin's  Fluxions,  Art.  828, 
&c.  Wood's  Algebra,  Art.  410,  Lacroix's  Comp.  Alg.  Art. 
81,  &:c.  Euler's  Anal.  Infin.  C.  xiii,  Simpson's  Essays  and 
Dissertations,  De  Moivre's  Miss.  Analyt.  p.  72,  and  the  Lon- 
don Philosophical  Transactions. 

Note  V.  p.  291. 

To  those  who  have  made  any  considerable  progress  in  the 
mathematics,  this  section  will  doubtless  appear  very  defec- 
tive. But  it  was  impossible  to  do  justice  to  the  subject, 
without  occupying  more  room  than  could  be  allotted  to  it 
here.  In  going  through  an  elementary  coirtse  of  mathema- 
tics and  natural  philosophy,  the  student  will  rarely  have  oc- 
casion to  solve  an  equation  above  the  second  degree. 

Those  who  wish  to  examine  particularly  the  different  meth- 
ods of- solution,  will  find  them  in  Newton's  Universal  Arith- 
metic, Maclaurin's  Alg.  Part.  2,  Euler's  Alg.  Part  1.  Sec.  4, 
Waring's  Algebra,  Do.  Medit.  Algeb.,  Wallis'  Algebra,  Simp- 
son's Alg.  Sec.  12,  Fenn's  Alg.  Ch.  3  and  4.,  Saunderson'a 
Alg.  Book  x,  Simpson's  Essays  and  Dissertations,  Journal 
De  Physique,  Mar.  1807,  and  the  Philosophical  Transactions. 

Note  W.  p.  298. 

It  w^ll  be  thought,  perhaps,  that  it  was  unnecessary  to  he 
BO  particular,  in  obtaining  the  expression  for  the  area  of  a 
parallelogram,  for  the  use  of  those  who  read  Playfair's  edi- 
tion of  Euclid,  in  which  ^^JID.DC  is  put  for  the  rectangle 
contained  by  AD  and  DC."  It  is  to  be  observed,  however, 
that  he  introduces  this,  merely  as  an  article  of  notation. 
(Book  II.  Def.  1.)     And  though  a  point  interposed  between 


332  ALGEBRA. 

the  letters,  is,  in  Algebra,  a  sign  of  multiplication ;  yet  he 
does  not  here  undertake  to  show  how  the  sides  of  a  parallelo- 
gram may  be  multiplied  together.  In  the  first  book  of  the 
Supplement,  he  has  indeed  demonstrated,  that  "  equiangular 
parallelograms  are  to  one  anotlier,  as  the  products  of  the 
liumbers  proportional  to  their  sides,"  But  he  has  not  given 
to  the  expressions  the  forms  most  convenient  for  the  suc- 
ceeding parts  of  this  work.  In  making  the  transition  from 
pure  geometry  to  algebraic  solutions  and  demonstrations,  it  is 
important  to  have  it  clearly  seen  that  the  geometrical  princi- 

f)les  are  not  altered ;  but  are  only  expressed  in  a  different 
anguage. 

Note  X.  p.  307. 

This  section  comprises  very  little  of  what  is  commonly 
understood  by  the  application  of  algebra  to  geometry.  The 
principal  object  has  been,  to  prepare  the  way  for  the  other 
parts  of  the  course,  by  stating  the  grounds  of  the  algebraic 
notation  of  geometrical  quantities,  and  rendering  it  familiar 
by  a  few  examples. 

On  the  construction  and  solution  of  problems.  See  New- 
ton's Arithmetic,  Simpson's  Alg.  Sec.  18  and  appendix,  La- 
croix's  App.  Alg.  Geom.,  Saunderson's  Alg.  Book  xiii,  Ana- 
lyt.  Inst,  of  Maria  Agnesi,  Book  i,  Sec.  2,  and  Emerson's 
Alg.  Book  II,  Sec.  6. 

Note  Y.  p.  320. 

On  the  equations  of  curves,  the  geometrical  construction 
of  equations,  the  finding  of  loci,  &c.  see  Maclaufin's  Alg. 
Part  III,  and  appendix,  Newton's  Arith.,  Emerson's  Alg. 
Book  II,  Sec.  9,  Do.  Prob.  of  Curves,  Euler's  Anal.  Infin., 
Waring's  Prob.  Alg.  and  Mansfield's  Essays. 

Among  the  subjects  which,  for  want  of  room,  are  entirely 
omitted  in  this  introduction,  one  of  the  most  interesting  is  the 
indeterminate  analysis.  No  part  of  Algebra,  perhaps,  is  bet 
ter  calculated  to  exercise  the  powers  of  invention.  But  other 
branches  of  the  mathematics  are  so  little  dependent  on  this, 
that  it  is  not  absolutely  necessary  to  give  it  a  place  in  an  ele- 
mentary course. 

See,  on  this  subject.  Eider's  Alg.  Vol.  ii,  with  Lagrange's 
additions,  Saunderson's  Alff.  Book  vi,  Bonnycastle's  Algebra, 
and  the  Edinburffh  Phil.  Transactions,  Vol.  ii. 


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